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CHAPTER 12 Combinatorics 529

CHAPTER CONTENTS

12A The addition and multiplication principles

12B Permutations

12C Factorials

12D Permutations using nPr

12E Permutations involving restrictions

12F Arrangements in a circle

12G Combinations using nCr

12H Applications to probability

CHAPTER 12

CombinatoricsDIGITAL DOCdoc-981310 Quick Questions

12A The addition and multiplicationprinciplesIntroductionConsider how many ways two different letters can be listed from the letters C, A and T if the order

in which the letters are to be listed is not taken into account. We would write CA, CT and AT. If

the order of listing the two letters is taken into consideration, there will be 6 different possibilities,

namely, CA, AC, CT, TC, AT and TA. In this chapter we introduce some methods that will enableus to effectively determine the number of possible ways objects can be ordered according to given

conditions, without necessarily having to list them. Combinatorial theory is widely applied in areas

such as computer system design, genetics, statistics and probability, where arrangements are of

particular importance.

The addition principleTo reach the top of the hill, Jack and Jill can use public transport (tram or bus) or private transport (hire

car, taxi or motorbike). In how many ways can Jack and Jill go up the hill if only one form of transport is

to be used for the entire trip?

Since the modes of transport are mutually exclusive (that is, two forms of transport cannot be used at

the same time), there are 2 +3 =5 different ways of travel.This straightforward method of summing is the addition principle, which can be stated as: If twooperations can be performed in A orB ways respectively, then both operations can be performed

together in A +B ways.

WORKED EXAMPLE 1

A particular mathematics problem can be solved in 2 ways using analytical methods, in 4 ways

using approximation techniques and in 3 ways by trial and error strategies. In how many ways

can the problem be solved?

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530 Maths Quest 11 Mathematical Methods CAS

THINK WRIT E

1 List the given information. Analytical 2

Approximation 4

Trial and error 3

2 Use the addition principle as the three methods

of solving the problem are mutually exclusive.The total number of ways is 2 +4 +3 =9.

WORKED EXAMPLE 2

A stack of playing cards contains four jacks, four queens and four kings. Gary has two jacks, a

queen and a king in his hand. But, to complete his hand, Gary requires three jacks, two queens

or two kings. In how many ways can he complete his hand?

THINK WRIT E

1 List the cards remaining in the stack. Two jacks, three queens and three kings remain inthe stack.

2 Gary requires a jack ora queen ora king tocomplete his hand. Use the addition principleto calculate how many ways he could completehis hand, given the cards that are remaining.

2 +3 +3 =8There are eight ways for Gary to complete his

hand.

The multiplication principleSuppose 4 colours are available to spray-paint 3 different cars. Let O1be the first operation selecting a car

and let O2be the second operation picking a paint colour. Also let C1, C2, C3and P1, P2, P3, P4denote,

respectively, the cars and the available colours. The 12 different ways in which the job can be undertaken are

{C1P1, C1P2, C1P3, C1P4, C2P1, C2P2, C2P3, C2P4, C3P1, C3P2, C3P3, C3P4}. Since the choice of a particular

car is independent of the colour selected, the total number of possibilities can be obtained by multiplying

together the number of choices available from the

two operations. That is, there are 3 4 =12 differentways possible. The tree diagram at right shows the

different outcomes.

The product of the number of outcomes from

each operation provides the total number of possible

outcomes of the operations performed sequentially.

This method is the basis of the multiplication

principle,which states: if two operations can be

performed in A and B ways respectively, then

both operations can be performed in succession in

A B ways.

WORKED EXAMPLE 3

Juanita has to choose an outfit to wear to a party. She has 6 skirts, 5 jumpers and 8 shirts to

choose from. If any combination of these items will be acceptable attire, in how many styles of

dress can Juanita attend the party?

THINK WRIT E

1 Choose a skirt, a jumper anda shirt. There are 6 skirts, 5 jumpers and 8 shirts.

2 Use the multiplication principle. The total number of ways is 6 5 8 =240.

P1P2P3P4

P1P2P3P4

P1P2P3P4

C1

C2

C3

Operation 1

O1

Operation 2

O2

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We can also represent the sequence of operations of the above example using boxed numbers as

follows.

Skirts Jumpers Shirts

6 5 8 =240

Each box contains the number of possible outcomes associated with the particular operation.

WORKED EXAMPLE 4

From a cafeteria 4-course lunch menu, I can choose 3 varieties of soup, 5 types

of seafood, 4 kinds of side dish and 2 types of salad.

a How many different dishes are offered?

b How many different lunches can be ordered if one dish from each course is

selected?

c How many different types of dish are possible if soup and seafood must be included

with each order?

THINK WR ITE

a Write the number of dishes for each course

and use the addition rule.

a The 4-course menu offers

3 +5 +4 +2 =14 different dishes.b There are 3 soups, 5 seafoods, 4 side dishes

and 2 salads. Use the multiplication rule

(as you must sequentially order a soup,

seafood, a side dish anda salad).

b Soup Seafood Side dish Salad

3 5 4 2 =120

Number of different lunches is 120

c 1 Consider the possible orders containing

soup and seafood:

soup and seafood only

c Soup Seafood

3 5 3 5 =15

soup and seafood and a side dish only Soup Seafood Side dish

3 5 4 3 5 4 =60

soup and seafood and a salad only Soup Seafood Salad3 5 2 3 5 2 =30

soup and seafood and a side dish and

a salad.

Soup Seafood Side dish Salad

3 5 4 2 3 5 4 2 =120

2 Calculate the number of dishes

possible for each order.

3 Use the addition rule to find the total

(as you can only order the first, second,

third orfourth combination).

Number of different types of dish possible

=15 +60 +30 +120=225

Exercise 12A The addition and multiplicationprinciples 1 WE1 Juicy Chickens offers 10 varieties of roast chicken dish, 6 types of fried chicken and 5 types of

chicken pie. How many different chicken meals are sold by Juicy Chickens?

2 Freda Frog eats 2 varieties of fly on the first day, 5 varieties on the second day, 9 varieties on the third

day and 14 varieties on the fourth day. Assuming Freda will never consume 2 of the same variety of fly

and that her daily eating habits follow this definite pattern, find how many flies she will eat altogether

in a week.

TUTORIALeles-1454Worked example 4

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3 A suburban mall consists of five shops: Teen Fashion, Harrys Takeaway, Video & Games Arcade, Toy

Palace and Byte Computers. On a busy weekend, 11 people went into Teen Fashion, 27 bought food

from Harrys Takeaway and 59 people entered the Toy Palace. Each person visited only one store. How

many customers did Teen Fashion, Harrys Takeaway and Toy Palace have altogether?

4 MC Two pieces of timber can be held together using adhesives, fasteners or clamps. The adhesives

are PVA glue, Liquid Nails and Bondcrete. Fasteners that can be used are nails, screws, rivets and bolts.

There are two different types of clamp available: SureGrip and Hold-tite. If only one adhesive fastener

or clamp is required, the number of ways two pieces of timber can be joined is:

A 2 B 24 C 3 D 4 E 9 5 WE2 From a pack of playing cards, the queen of spades, king of clubs and queen of clubs are drawn. In

how many ways can another card from the deck be drawn so that there will be three queens or two kings?

6 MC There are 4 novels, 7 comic books and 2 biographies on a bookshelf. Zoe selects and reads 2

novels, 3 comics and a biography from the shelf. However, her reading assignment requires that she

read 3 novels, 4 comics or 2 biographies. In how many ways can she select books from the shelf to

meet the minimum requirements of the assignment?A 6 B 7 C 11 D 13 E 24

7 WE3 Jack and Diane are preparing for their wedding. They will decide on one of 3 churches, one

of 5 available reception centres and one of 10 holiday destinations. How many combinations of church,

reception centre and holiday are possible?

8 Alana lives in Melbourne and intends to go to Sydney via Canberra. She will get to Canberra by bus,continue on to Sydney by hire car and return home by air. If 4 bus lines are available for the outward

journey to Canberra, 6 car rental agencies can be used to get from Canberra to Sydney and 3 airlines

are available for the return trip, determine how many different ways Alana can make the trip to Sydney

and back.

9 At Burpies restaurant the special meal consists of a choice of one of 2 entres, one of 3 main meat dishes

and one of 4 kinds of dessert. For a surprise feast at Belchies restaurant you can have one of 5 different

entres, select from 4 main meals and decide which one of 3 kinds of dessert to order.

a How many different combinations of dishes are possible in a special meal consisting of an

entre, a meat dish and a dessert?

b Find how many different combinations of dishes are available to a customer who visits both

places and orders a special meal and a surprise feast. (Assume that the customer must have an

entre, main meal and dessert for the surprise feast.)

10 MC On a dentists waiting room table are 3 piles of reading matter. The first pile consists of

6 different copies ofNews, the second pile has 5 different issues of Geographicand the third pile

comprises 10 different Womans Worldmagazines. A patient randomly chooses one item of reading

from each pile. The number of ways of choosing the 3 items is:A 21 B 30 C 216 D 19 E 300

11 MC A Whoppa pizza base is made using one of

2 types of cheese and one of 2 toppings. Up to 3

additional toppings are available at extra cost. The

number of different Whoppa pizzas that can be made

containing at least one additional topping is:A 12 B 16

C 24 D 28

E 2012 WE4 A school offers English, Maths, Language and

Science as part of the curriculum. Janice must do at

least one of these subjects.

a List the different ways Janice can select at least one

subject.

b In how many ways can this be done?

13 To get to school, Erin can walk, take the train or catch the bus. After school she can either walk or catch

the bus to get back home.

a List the different combinations of travel for Erin to get to school and to return to her home.

b Show the different travel methods as a tree diagram.

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14 A hot dog consists of a sausage in a bun with sauce. Onion, tomato, pineapple and cheese are

available as extras. How many different types of hot dog can be made?

15 During a special morning recess, teachers had a choice of tea, orange juice, coffee, pies, cheese,

salami, biscuits and cake. However, a teacher could sample only two kinds of food and one drink.

How many different combinations of two kinds of food and drink were possible?

12B PermutationsA permutation is the arrangement of objects in a specific order. Awarding a first and second prize to

two people randomly selected from a studio game-show audience or determining the number of ways

a group of people can queue for tickets are examples where the order of objects needs to be taken into

account.

Consider now how many ways two letters can be taken from the letters B, L, U and E and then

arranged.

If the order of the letters is taken into account and repetition of letters (that is, BB, LL etc.) is not

allowed, we have the 12 possible arrangements shown below:

BL, LB, BU, UB, BE, EB, LU, UL, LE, EL, UE, EU

We can obtain the same result using the multiplication principle. There are 4 choices for the first letter

because there are 4 letters available. Once the first letter has been chosen there are 3 letters to choose

from for the second letter.

First letter Second letter

4 3 =12

Notice that the multiplication principle takes into account the order of the outcomes. That is, BL is not

considered to be the same as LB, BU is not the same as UB and so on.

WORKED EXAMPLE 5

Josie picks up a Mathematics textbook, an English novel and a Biology notebook and places

them on a shelf. Determine the number of ways the books can be arranged. List the ways they

can be arranged.

THINK WRITE

1 There are three positions to be filled on the shelf. 3 2 1

2 There are three choices of book for the first position

on the shelf. This leaves two choices for the second

position and one choice for the third position.

3 Use the multiplication principle. 3 2 1 =6 arrangements

4 Let M be the Mathematics textbook, E the English

novel and B the Biology notebook.

The arrangements are MEB, MBE, BME,

BEM, EMB, EBM.

WORKED EXAMPLE 6

In how many ways can at least two letters be chosen from the word STAR if the order of the

letters is taken into account and repetition of letters is not allowed?

THINK WRITE

1 There are 3 mutually exclusive events: choose

2 letters from 4 letters, 3 letters from 4 letters, or

4 letters from 4 letters.

2 For the first event there are 4 choices for the first

letter and3 choices for the second letter, because

repetitions are not allowed.

2 letters 4 3 =12 ways

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3 For the second event there are 4 choices for the

first letter, 3 choices for the second letter and

2 choices for the third letter.

3 letters 4 3 2 =24 ways

4 For the third event there are 4 choices for the first

letter, 3 choices for the second letter, 2 choices for

the third letter and1 choice for the fourth letter.

4 letters 4 3 2 1 =24 ways

5 Use the addition rule to find the total numberof possibilities. Number of ways =12 +24 +24=60

WORKED EXAMPLE 7

How many ways are there for 2 different prizes or 3 different prizes to be

awarded to a group of 5 people if:

a a person may receive more than one award?

b a person may not receive more than one award?

THINK WRIT E

a 1 Use the multiplication principle to find the

number of ways for 2 prizes to be awarded.

Any one of the 5 people can receive the first

prize andany one of the 5 people can receive

the second prize because the same person

may receive more than one prize.

a 2 prizes

1st 2nd

5 5 =25

2 Use the multiplication principle to find the

number of ways for 3 prizes to be awarded

and remember that the same person may

receive more than one prize.

3 prizes

1st 2nd 3rd

5 5 5 =125

3 Use the addition rule to obtain the totalnumber of ways to distribute 2 or3 prizes.

Number of ways to distribute 2 or 3 prizes =25 +125 =150

b Use the same method as above, but repetition is

not allowed, so the number of people to choose

from is reduced each time.

b 2 prizes 5 4 =20

3 prizes 5 4 3 =60

Number of ways to distribute 2 or 3 prizes

=20 +60 =80

Exercise 12B Permutations 1 WE5 A chef restocks her collection of spices by placing jars of pepper, nutmeg, ginger and mint on

the shelf. In how many different ways can the 4 jars be placed in a straight line?

2 In how many ways can 6 students line up at the school canteen?

3 If there are 8 competitors in a race, in how many ways can the first three places be awarded?

4 To cancel an electronic alarm, a 5-digit code number must be entered into the code box. Assuming that

digits may be repeated, how many codes are possible?

5 Five items of mail are to be placed in 5 letterboxes. In how many ways can this be done if no

letterbox is to contain more than one item?


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6 A history quiz consists of matching 8 countries with their capital cities. In how many ways can a

contestant answer the quiz by randomly matching each country with a capital city?

7 How many ordered subsets consisting of two letters can be chosen from the word SUPERBLY if:

a a letter may be used more than once in each subset?

b choosing the same letter more than once is not permitted?

8 WE6 In how many ways can at least two letters be chosen from the word MATHS if the order of the

letters is taken into account and repetition of letters is not allowed?

9 WE7 Calculate the number of ways that 3 or 4 prizes can be awarded to a group of 5 people if:a a member of the group is allowed to receive more than one prize

b a member of the group cannot receive more than one prize.

10 Decide in how many ways 2 or 3 letters can be selected from the vowels of the alphabet if a vowel

can appear only once in each selection.

11 Determine how many numbers greater than 10 can be made using all of the digits 4, 7, 2, 6 and 5 if

each digit cannot be used more than once.

12 How many numbers greater than 100 and less than 10 000 may be formed using the digits 2, 3, 4 and 5

if each digit may be used more than once?

13 MC The number of 3-digit and 4-digit numbers greater than 500 that can be formed using the digits 2,

6, 1, 5 and 3, if each digit can be used more than once in each selection, is:A 600 B 500 C 675 D 575 E 450

14 MC Juliana has saved her pocket money to buy

up to 3 fashion magazines. If there are 5 different

magazines to choose from, the number of ways she

can buy 1, 2 or 3 magazines is:

A 90 B 80 C 25D 70 E 85

15 MC The total number of 2-digit, 3-digit and 4-digit

odd numbers that can be formed using the digits, 5, 3,

4 and 9, when a digit may be used more than once in a

group, is:A 78 B 85 C 252 D 68 E 75

16 How many 7-letter arrangements are possible using the letters of the word DECAGON if the letters A,

E and O must occupy the third, fifth and sixth positions respectively and the letters remaining may beused more than once?

17 A school fundraising competition that costs 5 cents per entry involves trying to correctly match

9 teachers with their baby photographs. Wasim wants to be certain to win the $1000 first prize by trying

all possible combinations. Decide how much money Wasim will win or lose if he is to be the prize

winner.

18 A version of the party game musical chairs has the players march around a line of chairs and scramble

to sit on them when the music unexpectedly stops. At each stage the number of players is one more

than the number of chairs. The player who remains standing when the music stops is out of the game

and one chair is then removed. The player remaining sitting after all the other players have been

eliminated is the winner.

a If 12 players are taking part, how many different arrangements of seating are possible during the:

i first round? ii fourth round?b The rules are changed so that 2 chairs are removed each time. If there are 9 players and 7 chairs at

the start of the game, how many seating arrangements are possible for all the rounds?

19 A school is using identification cards (ID cards) that consist of 3 letters selected from A to E inclusive

followed by 3 digits chosen from 0 to 9 inclusive.

a How many different ID cards can be issued to students if a digit may be used more than once but

all 3 letters of each ID are different?

b New ID cards are issued to all students each year and the old cards discarded. However, the old

ID numbers are not used again. If, on average, the schools population increases by 10% each

year and was 2000 during the year when the ID cards were first used, how many years will elapse

before cards with numbers already used will have to be issued?

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12C FactorialsExpressions obtained by using the multiplication principle frequently contain the product of consecutive

whole numbers. It is convenient to adopt a shorthand way of representing such expressions to assist with

calculations and to effectively display the properties associated with permutations and other types of

order of objects. Particularly useful is to define n! to mean the product of nconsecutive positive integers

starting from ndown to 1. That is:

n! =n(n1) (n2) (n3) . . . 32 1

The symbol n! is read as nfactorial.

For example, 4! =4 3 2 1 =24, 3! =3 21 =6, 2! =2 1=2.Alternatively, 4! =4 3! =24, 3! =3 2! =6, 2! =2 1! =2.

Thus from the definition we have n! =n(n1)! orn

nn

!

( 1)!=

If we substitute n=1 we have:1!

(1 1)!1

= or1

(0)!1= (since 1! =1)

This expression is true if 0! is taken to be equivalent to 1. So we define 0! =1.

WORKED EXAMPLE 8

a Express 7! as a numeral. b Simplify 2 5! +(3 2)!

THINK WR ITE

a 1 Use the definition of n! with n=7. a 7! =7 6 5 4 3 2 1

2 Multiply the numbers in the expression obtained. =5040

b 1 Write the expression and simplify (3 2)! b 2 5! +(3 2)! =2 5! +6!

2 Calculate 5! and 6! =2 120 +720

3 Evaluate. =960

WORKED EXAMPLE 9

Simplify8!

3!

THINK WR ITE

Divide the answer for 8! by the answer for 3!

using a calculator.

8!

3!

8 7 6 5 4 3!

3!=

=8 7 6 5 4=6720

WORKED EXAMPLE 10

a Evaluate 8! b Simplify100!

98!

THINK WR ITE

a 1 Use the factorial (!) feature of a CAS calculator

to evaluate the expression.

a 8!

2 Record the result. 40 320

3 State the answer. 8! =40 320

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b 1 100! and 98! are too large to write in fully

expanded form.

b

2 Express 100! with 98! as a factor.

100! =100 99 (98 . . . 3 2 1)=100 99 98!

100!

98!

100 99 98!

98!=

3 Cancel 98! in the expression. =100 99=9900

Notice that: n! =n(n1)!=n(n1) (n2)!=n(n1) (n2) (n3)! etc.

Exercise 12C Factorials 1 WE8a Evaluate:

a 4! b 9! c 12! d 3! + 2!e 5! 4! f 7! 6! 2! g 6! (1! +2! +3! +4! +5!)

2 WE8b Evaluate:

a 4 3! 4! b (4 +2!) 3! +5! c 5 6! 6 5!d 7 7! (8! 7!) e 8! +3 2! 5! f 7 9! +3 3! 9 8!g (5! 4!) +(8! 7!) h 12! +6! 11! 3 4!

3 MC The value of2! 3! 4!

2! 3!

+ +

+

is:

A 5 B 4 C 7 D 24 E 8

4 MC The value of 4(4! 3) +2!(5! 4!) is:A 250 B 235 C 284 D 276 E 290

5 WE9 Simplify:

a4!

2!b

5!

4!c

7!

3!d

6!

3!e

3!

2!

1!

0!+

6 WE10 Simplify:

a102!

100!b

1000!

998!c

4500!

4499!

d250!

247!e

396!

393!f

25000!

24999!

7 Simplify:

a15!

14!b

28!

26!c

55!

53!

d1000!

998!e

63!

(936 875)!

8 MC(12 37)!

(100 53)!

+

is equal to:

A 3250 B 1875 C 2840 D 1030 E 2352

9 Evaluate each expression.

a7!

4!

4!

3!+ b

9!

7!

32!

31!+ c

13!

10!

6!

3!

d80!

77!

10!

6!

64!

62! + e

8!

6!

12!

11!

78!

77! +

10 Simplify each expression.

a2!5!

3!b

6!

2! 3!+c

7! 5!

5!

d8! 4!

2! 3!

2!

3!

+

+

e18!4!

17!5!

8!10!

9!9!

DIGITAL DOCdoc-9814WorkSHEET 12.1

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12D Permutations using nPrA permutation is an arrangement of objects in which order is important. Consider the letters A, B and C.

There are 6 possible arrangements or permutations of these three letters:

ABC ACB BAC BCA CAB CBA

We could have determined that there are 6 possible arrangements without listing all of them using

the multiplication principle, where each box below represents a position (first letter, second letter, third

letter):

1st

letter

2nd

letter

3rd

letter

3 2 1 =6 ways

Note that we had 3 possibilities for the first letter but, having placed it, we were left with

2 possibilities for the second letter and, in turn, just 1 possibility for the third.

But what if we had 10 different letters and wished to select an arrangement of 3 letters? Again, we

could count the number of arrangements as follows:

1st

letter

2nd

letter

3rd

letter

10 9 8 =720 ways

We can express the above calculation using factorials as follows:

10 9 8 10 9 8 7 6 5 4 3 2 1

7 6 5 4 3 2 1 =

10!

7!=

10!

(10 3)!=

Following on from this, we can generalise a formula for the number of arrangements (permutations)

of nobjects, taking rat a time, which we denote by nPr:

=

n

n r

P !

( )!

n

r, where nand rare natural (counting) numbers, and rn.

Another way of thinking of nPris as n! expanded to rplaces.

= +n n n n r

r

P ( 1)( 2). . .( 1)

valuesmultiplied together

n

r

In the preceding example, which involved arranging 10 (n=10) objects (letters) taking 3 (r=3) at atime, we can verify that (nr+1) =(10 3 +1) =8, which was the last value in the chain of multipliednumbers.

Special cases1. If r=0, then P Pnr

n

0=

n

n

!

!=

=1This implies that there is one way of selecting zero objects from nobjects.

2. If r=n, then P Pn rn

n=

n!

0!=

=n!There are n! ways of arranging nobjects taken from nobjects.

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WORKED EXAMPLE 11

Calculate 7P3.

THINK WRIT E

Evaluate using the definition nPr. P 7!

(7 3)!

73 =

7!

4! or

7 6 5 4!

4!= =

5040

24 7 6 5= =

= 210 =210

WORKED EXAMPLE 12

Only the 3 fastest cars in a car rally of 10 competitors will compete in the world championships.

How many different arrangements of the 3 fastest rally cars are possible?

THINK WRIT E

1 We want the number of permutations when3 objects are selected from 10 objects.

2 Use nPrwith n=10 and r=3. Number of arrangements=10P3=10 9 8 =720

3 Alternatively, use the permutations feature of

the CAS calculator with n=10 and r=3.

nPr (10, 3)

4 Record the result. 720

WORKED EXAMPLE 13

How many numbers greater than 1000 can be formed using the digits 4, 5, 6, 7, 8 and 9 if a digit

cannot be used more than once?

THINK WRIT E

1 Each 4-digit, 5-digit or6-digit number formed

will be greater than 1000.

2 Find the number of ways the required number

of digits can be chosen from the 6 digits given.

6P4+6P5+

6P6

3 Add the 3 answers. (or situation). =360 +720 +720=1800There are 1800 numbers greater than 1000 that

can be formed.

WORKED EXAMPLE 14

A captain and vice-captain are to be chosen from a group consisting of 10 cricket

players. From the remaining 8 players, 3 will be selected to be the wicket keeper,

spin bowler and fast bowler. Calculate how many different ways the

5 positions can be allocated.

THINK WRIT E

1 Find the number of ways in which 2 objects

(captain/vice-captain) can be chosen from

10 objects (10 cricket players).

10P2


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2 Find the number of ways in which 3 objects

(wicket keeper/spin bowler/fast bowler) can

be chosen from 8 objects (8 remaining cricket

players).

Number of different ways

=10P28P3

=90 336

3 Multiply the two results (and situation as we

wish to have a captain and vice captain anda

wicket keeper, spin bowler and fast bowler).

=30 240

Exercise 12D Permutations using nPr 1 WE11 Evaluate:

a 6P4 b 8P2

c 9P3 d 4P4

e 25P5 f 3P2

g 4P2+5P1 h

8P67P3

i 6P35P4 j

3P14P2

K 100P4 l 200P3

2 WE12 A committee comprising a president, vice-president, secretary and treasurer is to be selected

from a group of 25 people. How many different committees are possible? 3 In how many ways can a first and second prize be given to 5 lottery winners?

4 WE13 How many numbers greater than 100 can be formed from the digits 2, 3, 5, 7 and 9 if a digit

cannot be used more than once?

5 John has a 5-cent coin, a 20-cent coin, a 50-cent coin and a $2 coin.

a In how many ways can the coins be placed in a row?

b In how many ways can 2 coins or 3 coins be chosen if the order is taken into account?

6 MC A magic paint set contains seven magic colours that when applied to paper produce other

colours. The colour obtained depends on the order in which the colours are applied, and at least

two colours must be used. The number of different colours that can be produced is:

A 11 605 B 10 254 C 14 250D 12 540 E 13 692

7 WE14 A captain and vice-captain are to be selected from a team of 18 footballers. From the remaining

16, four players will be selected to be the full-back, full-forward, centre-half back and centre-half

forward. Calculate the number of ways the 6 positions can be allocated.

8 The Southern Belles train crew consists of 2 drivers and 4 engineers. Each person performs different

tasks. The 2 drivers are chosen from 6 available drivers and the 4 engineers from 10 engineers. How

many permutations of the trains crew are possible?

9 Three students are to be chosen from a group of 8 students to fill the positions of school president,

vice-president and treasurer. After these appointments are made, 2 more students will be selected from

the group to serve as secretary and assistant secretary. Determine how many different committees are

possible.

10 A novelty sports day carnival involves

10 competitors. A prize is given to the winner

of the first race, who then cannot take part in

the remainder of the races. The winner and

runner-up of the second race are awarded

prizes and are then eliminated from the

remainder of the events.

Similarly, the first three place-getters of the

third race are given prizes and must drop out

of the competition. This is continued until the

number of competitors remaining is the same

as the number of prizes to be awarded. How

many different ways can prizes be awarded?

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SkillSHEET 12.1Calculating nP

r

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11 There are three separate bundles of reading material comprising 4 comics, 2 novels and 3 magazines.

They are placed together to form one pile.

a In how many ways can this be done if there are no restrictions on where individual items are to be

placed?

b Determine the number of permutations if

the order of the comic books in each bundle

does not change.

12E Permutations involving restrictionsIdentical objectsSo far our study of permutations has been based on the assumption that the objects arranged were all

different (distinguishable). We will now examine the situation when some of the objects are identical

(indistinguishable).

A scrabble player has the following letter tiles: A, A, A, B, C, D, E. If the As were distinguishable, we

might consider them to be A1, A2, A3and could begin to list the possible arrangements of the 7 letters as

follows:

A1A2A3BCDE ....... ....... ....... and so onA1A3A2BCDE ....... ....... ....... and so on

A2A1A3BCDE ....... ....... ....... and so on

A2A3A1BCDE ....... ....... ....... and so on

A3A1A2BCDE ....... ....... ....... and so on

A3A2A1BCDE ....... ....... ....... and so on

Without listing them all, we can calculate there are 7P7=7! =5040 possible arrangements. But the Asare not distinguishable. So, really, the arrangements listed above are all the same as AAABCDE, which

counts as one arrangement.

Because there are 3 As we have 3! =6 times too many arrangements, hence we need to divide5040 by 6.

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This means there are only = = =P

3!

7!

3!

5040

6840

77 different arrangements or permutations of

7 objects where 3 of them are identical.

In general, the number of arrangements ofnobjects,pof which are identical, is given byn

p

!

!

Extending this formula we have:

The number of ways of arrangingnobjects that includep identical objects of one type,

q identical objects of another type, r identical objects of yet another type and so on is:

n

p q r

!

! ! ! . . .

WORKED EXAMPLE 15

In how many ways can 4 identical red marbles and 3 identical blue marbles be placed in a row?

THINK WRIT E

1 There are 4 +3 = 7 objects altogether.

2 The number of ways the blue marbles can be

arranged is 3!, and the number of ways the red

marbles can be arranged is 4!.

3 Substitute the values into the formula. Number of ways7!

4! 3!

35

=

=

Grouped objectsIn how many ways can the letters A, B, C, D be positioned in a row? We know that this can be done in

4! ways, but what would be the answer if the question had been: In how many ways can the letters A,

B, C, D be positioned in a row if A and B must be next to each other? The number of arrangements willclearly be less than 4! because of the restriction imposed on A and B. The figure below shows the 4!

possible arrangements of A, B, C, D that include the 12 ways A and B are together.

A B C D B A C D C B A D D B C A

A B D C B A D C C B D A D B A C

A C B D B C A D C A B D D C B A

A C D B B C D A C A D B D C A B

A D B C B D A C C D B A D A B C

A D C B B D C A C D A B D A C B

If A and B are to be together, we consider the problem to be one of arranging 3 objects, say X, C

and D, where one of the objects, X, is the group containing A and B.

The figure below shows that there are 6 arrangements with A and B together.A B C D C A B D C D A B

A B D C D A B C D C A B

The 3 objects can be arranged in 3! ways, and within the group A and B can themselves be arranged

in 2! ways (namely AB and BA). The multiplication principle is now used so that the number of

arrangements when A and B are together is 3! 2! =12.Now consider the permutations if A, B, C must be together. Again, we view the letters as consisting of

two objects, X and D, where X is the group of letters A, B and C. Thus we have two objects to arrange in

2! ways as shown below.

X D D X

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Among themselves the letters A, B, C contained in X have 3! different arrangements as shown below.

A B C D D A B C

A C B D D A C B

B A C D D B A C

B C A D D B C A

C A B D D C A B

C B A D D C B A

Therefore the total number of arrangements when A, B and C are together is 2! 3! =12.We can generalise this approach to include any number of groups of objects that are required to be

together.

Ifnobjects are to be divided intomgroups with each group having G1, G2, G3, . . . Gm

objects respectively, the number of arrangements is given by m! G1! G2! G3! . . . Gm!

WORKED EXAMPLE 16

The letters of the word TABLES are placed in a row. How many arrangements are possible if the

letters T, A and B must be together?

THINK WRIT E

1 Consider the letters T, A and B as one object

(group). There are 4 objects to be arranged,namely the TAB group and the letters L, E

and S.

G1={T, A, B}G2={L}G3={E}G4={S}m=42 Identify mand G1, G2, G3, G4.

3 Apply the formula:m! G1! G2! G3! G4!

Number of arrangements

=m! 3! 1! 1! 1!=4! 3!=144

WORKED EXAMPLE 17

Five cars a Toyota, a Ford, a Holden, a Mazda and a BMW are to be parked side by side. In

how many ways can this be done if the Toyota and BMW are not to be parked next to each other?

THINK WRIT E

1 The five cars can be arranged in 5! ways

without restriction.

Number of ways of arranging 5 cars

=5!

2 Calculate the number of arrangements where

the Toyota and BMW are together (4! 2!).(m=4, G1=2, G2=1, G3=1, G4=1)

Number of ways where the Toyota and BMW are

not together

=5! 4! 2!

3 Subtract from the unrestricted number of

arrangements the number of ways the two cars

are together.

=120 48 =72

WORKED EXAMPLE 18

The letters of the word REPLETE are arranged in a row. In how many

ways can this be done if the letters R and P must not be together?

THINK WRIT E

1 Find the number of unrestricted arrangements

of the 7 letters and consider that there are

3 identical Es.

Number of ways of arranging 7 letters with 3 Es

=7!

3!


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2 Calculate the (restricted) number of ways R

and P are together. Consider R and P as one

object so there are 6 objects to arrange. There

are three Es to consider (3! ways). R and P can

be arranged in 2! ways within their group.

(m=6, G1=2, G2=G3=G4=G5=G6=1)

Number of ways where R and P are not together

= 7!

3!

6! 2!

3!

3 Subtract the number of ways with R

and P together from the total number of

arrangements.

=840 240

=600

Exercise 12E Permutations involving restrictions 1 WE15 In how many ways can 5 identical white beads and 4 identical yellow beads be arranged in a

straight line?

2 Three 5-cent coins, two 10-cent coins and six 20-cent coins are to be placed side by side. Determine

how many ways this can be done.

3 MC The number of permutations using the letters of the word LOOPHOLE is:

A 3520 B 3360 C 4000D 4150 E 3840

4 The toy set shown in the photo consists of a number of

indistinguishable brown horses, 1 white horse, a cowboy

and 3 indistinguishable black horses. In how many

ways can they be placed end to end?

5 How many different 6-digit numbers can be obtained

using the digits 4, 6, 7, 6, 6 and 4?

6 A party-light kit consists of 20 coloured globes connected to

each other in a straight line.

a If there are 5 red globes, 6 blue globes, 7 yellow globes

and a number of green globes as shown at right, find

how many different arrangements of coloured globes

are possible.b How many different permutations of coloured globes

are there if the first and last globes must both be red?

7 WE16 Find how many arrangements are possible altogether when

the letters of the word CHAIR are placed in a row and C and H are to

be next to each other.

8 The digits 5, 3, 6, 2 and 7 are used to make a 5-digit number. How many different numbers are

possible if the digits 3, 2 and 7 must be together?

9 Maria, Steven, James, Sofia, Nin and Alfredo are standing next to each other. Calculate how many ways

this can be done if Maria and James are not to stand next to each other.

10 WE17 Establish the number of ways in which 7 different books can be placed on a bookshelf if 2

particular books must occupy the end positions and 3 of the remaining books are not to be placed together.

11 MC Ten athletes line up for a race. The number ofpermutations when three of the athletes Sam,

Troy and Pablo would be next to each other is:

A 3 628 800 B 1 209 600 C 241 920

D 5 443 200 E 4 838 400

12 A carpenter has 3 identical hammers, 5 different

screwdrivers, 2 identical mallets, 2 different saws and

a tape-measure. She wishes to hang the tools in a row

on a tool rack on the wall. In how many ways can this

be done if the first and last positions on the rack are to

be mallets and the hammers are not to be all together?

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13 WE18 Decide in how many ways the letters of the word ABRACADABRA can be arranged in a row

if C, R and D are not to be together.

14 MC The number of ways the letters of the alphabet can be placed in a straight line with the restriction

that the letters of the sentence UP THE BIG SKY WORLD must not be together is:

A 26! 11!16! B 26! +16! C 16! 8! D 6!16! E 6!16!26!

12F Arrangements in a circleAnna, Betty and Lin stand on the circumference of a circle painted on the schools playground. In how

many different arrangements can the three girls stand?

The figure below shows the two arrangements for the girls positions on the circle.

Betty

Lin Lin Betty

AnnaAnna

Notice that Anna is locked in position to provide a reference point, and Betty and Lin are arrangedaround Anna in 2! (=2) ways.

Compare this with the 3! (=6) arrangements in a line.

ABL BAL BLA LBA ALB LAB

(A is Anna, B is Betty, L is Lin)

Susie now joins the group to make 4 people in a circle.

We can designate any of the 4 girls in the circle as our start by fixing one person (in this case,

Anna) in one position and arranging the remaining girls around her. This reduces, by one person, the

number of girls to arrange.

(Ais Anna, Bis Betty, Lis Lin, Sis Susie)

A

B L

S

A

B S

L

A

L B

S

A

L S

B

A

B L

S

A

S B

L

There are 3! (=6) ways of arranging 4 people in a circle. Compare this with 4! (=24) arrangements ina line.

In general:

ndistinguishable objects can be arranged in a circle in (n 1)! ways.

In how

many ways

can these

five children

be arranged

in a circle?

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WORKED EXAMPLE 19

In how many ways can the vowels of the alphabet be arranged in a circle?

THINK WRIT E

1 The vowels are a, e, i, o, u. Therefore, there are

5 objects to arrange.n=5

2 Use (n1)! with n=5. Number of ways =(n1)!=(5 1)!=4!=24

WORKED EXAMPLE 20

Calculate the number of arrangements in a circle that are possible using the letters of the word

UNUSUALLY.

THINK WRIT E

1 There are 9 letters, so use n=9 with (n1)! n=9

2 We need to consider repetition of letters.

There are three Us and two Ls.

Number of arrangements =(9 1)!

3!2!

=

=

8!

3! 2!

3360

WORKED EXAMPLE 21

In how many ways can 6 people sit around a table if two particular people must

be seated next to each other?

THINK WRIT E

1 Consider the two people required to sit

together as being one object. So there are

5 objects to arrange in a circle.

n=5

2 The two people can be arranged in 2! ways.

3 Use the multiplication principle. Number of ways =(5 1)!2!=4! 2!=48

Exercise 12F Arrangements in a circle 1 WE19 Calculate the number of ways in which the letters of the word PENCIL can be arranged in a

circle.

2 MC Eight children hold hands to form a circle in the playground. The number of ways this can be

done is:

A 6280 B 5400 C 3680 D 4320 E 5040

3 WE20 Determine the number of arrangements in a circle that are possible when the letters of the word

EXCELLENT are used.

4 A child uses coloured dots on paper to represent the hour marks of a clock face. How many permutations

are possible if there are 4 orange dots, 5 white dots, 2 black dots and 1 purple dot?

TUTORIALeles-1458

Worked example 21

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5 WE21 A family of 3 adults, 3 boys and 3 girls are sitting around a circular dinner table. Find the

number of seating positions that are possible if the 3 boys are to be together.

6 A special pizza consists of 10 slices with different toppings used. If 2 slices are Capricciosa, 5 slices are

Supreme and 3 slices are Ham and Pineapple, how many different arrangements of pizza slices are possible?

7 A manufacturer of merry-go-rounds uses 8 identical wooden horses, 4 identical plastic motorbikes and

2 different miniature cars. They are all equally connected around the rim of a circular moving base.

Establish how many different arrangements there can be if the 2 cars are not to be placed in consecutive

positions.

8 MC Ten owners of pedigree dogs will enter the arena to parade their dogs by walking around a

circular track. Unfortunately, 3 particular dogs cannot get along together and so cannot parade if all

3 are next to each other. There appears to be no problem if any two of this group of 3 dogs are together.

The number of ways of avoiding this problem is:A 358 848 B 387 072 C 362 880 D 332 640 E 354 065

9 In how many ways can the letters of the word POTATOES be arranged in a circle?

10 MC The letters of the word FULFILLED are to be arranged in a circle. The number of arrangements

possible when U and E are together or when U, E and D are together is:A 3140 B 1940 C 2000 D 1200 E 1850

11 To publicise a venue, a hotel manager gave a gift to each of 12 prominent businesspeople as they went

into the conference room and seated themselves at a round table to begin discussions. The gifts comprised

4 fountain pens, 5 pocket electronic organisers and 3 calculators. Calculate what fraction of the possible

unrestricted arrangements is the number of arrangements that has 4 businesspeople who have been given

a fountain pen sitting next to each other.

12G Combinations using nCr

Taking combinations involves the selection ofrobjects from nobjects without consideration for the

order of the elements. For example, the number of permutations of two letters selected from the letters

A, B, C, D is 4P2=12. The arrangements are:

AB AC AD BC BD CD

BA CA DA CB DB DC

If we are not concerned with order, there are only 6 selections:

AB AC AD BC BD CD

The 2! ways of arranging the elements of the 2-element subgroup are not considered.

Now consider the selection of 3 letters from A, B, C, D. The number of ordered subsets is 4P3, and

each subset of 3 elements can be arranged in 3! ways. Therefore 4P3is the number of unordered subsets

of 3 objects multiplied by the number of ways the 3 objects can be arranged.

In general terms it can be stated that nPris the number (nCr) of unordered groups of robjects

multiplied by the number of arrangements (r!) of robjects.

That is, nPr=nCrr! so that

nCr =r

P

!.

n

r

Now by the definition of =

n

n r

P !

( )!

n

rwe have:

=

=

n

n rr

n

r n rC

!

( )! !

!

!( )!

n

r

The number of combinations is usually denoted by

n

r

C orn

r, so we have:

1. The number of combinations ofrobjects selected fromnobjects is:

= =

n

r

n

r n r

C !

!( )!

n

r

wheren,rare natural numbers andrn.

2. =

= =r nr r

P C ! or C P

!

n

r

n

r

n

r

n

r

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The functionnCris a standard mathematical function to be found on scientific, graphics and CAS

calculators.

Special cases1. If r=0, then = =

=

=n

n

n

n

C C !

0! ( 0)!

!

1 !1.n

r

n

0

This implies that there is one way of selecting 0 objects from nobjects.

2. If r=n, then= =

=

=n

n n n

n

nC C

!

! ( )!

!

! 0! 1.

n

r

n

n

There is one combination of nobjects taken from nobjects.

3. If r=1, then = =

=

=n

n

n n

n

nC C !

1! ( 1)!

( 1)!

1 ( 1)!.n

r

n

1

If objects are taken one at a time from nobjects, there are ncombinations.

From cases 1 and 2 we conclude that nC0=nCn.

This is an instance of the general case that:

=

=

n

r

n

n r

C C ornr

n

n r

For example, =C 7!

4!3!

74

and =C

7!

3!4!

7

3

so 7C4=7C3

WORKED EXAMPLE 22

Evaluate 10C3.

THINK WRIT E

1 Use the definitionn

n n r

C !

!( )!.n

r =

=

=

=

=

C 10!

3!(10 3)!

10!

3! 7!

10 9 8 7!

3! 7!

10 9 8

3 2 1

103

=1202 Alternatively, use the combinations feature of

the CAS calculator with n=10 and r=3.

nCr (10, 3)

3 Record the result. 120

WORKED EXAMPLE 23

Evaluate100

98

.

THINK WRIT E

1 Express100

98

in factorial form. 100

98

100!

98!2!

100 99 98!

98! 2 1

=

=

2 100! =100 9998!

3 Evaluate. 100 99

2 1

4950

=

=

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WORKED EXAMPLE 24

In how many ways can a committee of 2 boys and 3 girls be formed from a group consisting of

5 boys and 8 girls?

THINK WRIT E

1 Select 2 boys from 5 boys.

2 Select 3 girls from 8 girls.

3 Use the multiplication principle (and

situation).Number of ways=5C2

8C3=10 56=560

WORKED EXAMPLE 25

A committee of 6 is to be formed from a group of 5 men and 4 women.

a How many committees can be formed?

b How many committees contain 3 men and 3 women?

c How many committees contain at least 4 men?

THINK WRIT E

a Use n=9 and r=6 with nCr. a Number of committees =9C6

=84

b 1 Select 3 men from 5 men and 3 women

from 4 women.

b Number of committees

=5C34C3

2 Use the multiplication principle.

(and situation)=10 4=40

c 1 At least 4 men means 4 men and 2 women

or 5 men and 1 woman.

c Number of committees

=5C44C2+

5C54C1

2 Select 4 men and 2 women. =5 6 +1 4

3 Select 5 men and 1 woman. =30 +4

4 Sum the answers because the 2 events are

mutually exclusive (or situation).=34

Exercise 12G Combinations using nCr 1 WE22 Calculate each of the following.

a 5C2 b 4C3 c

6C1 d 8C0 e

9C9

2 Evaluate each of the following.

a 6

4

b 7

5

c 10

2

d9

3

e 12

6

3 WE23 Determine the value of each of the following.

a30

29

b

55

53

c

64

61

d

38

34

e 29

24

4 MC The value of 2 4C2+3 5C3is:

A 42 B 90 C 80 D 94 E 70

5 Calculate each of the following.a 3C1and

3C2 b4C1and

4C3 c5C2and

5C3 d9C3and

9C6


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6 Copy and complete the following.

a 20C7=20C__ b 100C9=

100C__

7 In how many ways can 5 objects be chosen from 12?

8 How many combinations are possible if 2 numbers are chosen from 6 in a mini-lotto game?

9 A student must choose 5 types of party food from the following list: sausage rolls, potato crisps, fairy

bread, party pies, cheezels, cocktail frankfurts and celery sticks. How many different combinations may

be chosen?

10 A committee of 6 must be chosen from a meeting of 30 people. How many different committees arepossible?

11 WE24 In how many ways can a group of 3 boys and 4 girls be formed from a group consisting of

4 boys and 6 girls?

12 A magazine pile in a waiting room contains 6 glamour magazines and 7 computer magazines. In how

many ways can a patient choose 2 glamour and 3 computer magazines to flick through during a lengthy

wait?

13 A school offers 10 science subjects and 15 humanities subjects to prospective Year 12 students. In how

many ways may a student choose 4 science and 2 humanities subjects?

14 How many 10-card hands containing exactly 7 hearts and 3 spades are possible from a standard 52-card

deck?

15 WE25 A committee of 5 parents is to be established from a group of 6 men and 4 women.a Find how many different committees can be formed.

b How many different committees are possible consisting of 3 men and 2 women?

16 A school organises an adventure camp for its Year 11 students, who must choose 2 or 3 activities

from the following: paragliding, abseiling, skydiving and bungee jumping. In how many ways may a

group of activities be chosen?

17 An ice-cream vendor offers chocolate, strawberry and vanilla ice-creams with one, two or three scoops.

How many different ice-creams are possible? (Assume that you cannot choose two scoops of the same

flavour for any one one ice-cream.)

18 A basketball squad of 10 must be chosen from a group of 8 women and 6 men. How many squads are

possible:

a without restriction?

b if the squad contains 6 women and 4 men?c if the squad must contain at least 6 women?

d if the squad contains all the men?

19 A sub-committee of 3 people must be chosen from a group of 9 teachers (which includes the

principal). How many sub-committees may be chosen:

a that contain the principal?

b that do not contain the principal?

20 To win LottoMania, the 5 numbers entered on the players entry ticket must be the same as 5 numbers

that are randomly selected from the numbers 1 to 30.

a How many different entries are possible?

b What is the percentage increase in the number of possible combinations if the numbers are

randomly selected from the numbers 1 to 35?

21 MC A painter has 7 colours at her disposal. The number of additional colours that can be obtained by

mixing equal amounts of any number of the 7 colours is:

A 100 B 128 C 5040D 5120 E 120

22 Determine the number of ways in which 8 people can be divided into

2 equal groups.

23 MC The number of ways in which 10 objects can be divided into

2 unequal groups is:

A 385 B 835 C 950D 640 E 565

DIGITAL DOCdoc-9818

SkillSHEET 12.2Listing possibilities

DIGITAL DOCdoc-9819

InvestigationPascals triangle

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12H Applications to probabilityWe define the probability of an event to be:

Pr(event) number of favourable outcomes

total number of possible outcomes=

The methods we have used to calculate permutations and combinations can also be applied to

problems involving probability.

WORKED EXAMPLE 26

Romina makes a guess as to which 2 of 10 swimmers will come first and second in a race. What is

the probability that her guess will be right?

THINK WRIT E

1 Calculate in how many ways 2 swimmers can

be chosen from 10 swimmers, where the order

is taken into account. Use nPrwhere n=10 andr=2.

2 Use the formula for probability. The number

of favourable outcomes is 1 because Romina

makes only one guess.

=

=

Pr(correct guess) 1

P10 21

90

WORKED EXAMPLE 27

A computer randomly interchanged the letters of the word CREATIONS. Find the probability

that the letters A and T end up together.

THINK WRIT E

1 If A and T are together, treat them as one

object; therefore, we have 8 objects

2 AT can be arranged in 2! ways.

3 Use the formula for probability to find the

number of ways the 9 letters can be arranged

(total number of possible outcomes).

Pr (A and T are together)8! 2!

9!=

= 29

WORKED EXAMPLE 28

A committee of 5 people is to be formed by choosing members from a group of 6 men and

4 women. What is the probability that the committee will consist of 3 men and 2 women?

THINK WRIT E

1 Calculate the number of ways in which 3 men

can be selected from 6 men and 2 women can be

chosen from 4 women.

6C3and4C2

2 Use the multiplication principle to establish

the number of favourable outcomes of the

committees consisting of 6 men and 4 women.

6C34C2

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3 Using the formula for probability, determine

the number of ways in which 5 people can be

selected from the group of 10 people (total

number of possible outcomes).

Pr (3 men and 2 women)

C C

C

63

42

105

=

20 6

252

10

21

=

=

WORKED EXAMPLE 29

Eight people randomly seat themselves about a circular table. What is the probability that

3 particular people will be sitting next to each other?

THINK WRIT E

1 Treat the 3 people as one object; therefore,

there are 6 objects to arrange.

3!

2 Use the formula (n1)! for arrangements in a

circle for the situation where the 3 people aretogether.

(6 1)! 3!

3 Using the formula for probability, calculate

the total number of possible outcomes

for the 8 people, using (n1)!

Pr (3 particular people seated together)

(6 1)! 3!

(8 1)!

5! 3!

7!

720

5040

=

=

=

=0.143

WORKED EXAMPLE 30

Two bags (A and A) contain blue marbles (B) and other coloured marbles (B).A bag is randomly selected, then from that bag a marble is randomly selected.

The table below describes the distribution of marbles between the bags.

Bag A Bag A

5 blue marbles 4 blue marbles

3 other marbles 6 other marbles

a What is the probability of choosing bag A and then a blue marble?b What is the probability of not choosing bag A and then obtaining a blue marble?

c What is the probability of choosing a blue marble?

THINK WRIT E

a 1 Find the probability of choosing bag A. a =

=

=

Pr(A)number of favourable outcomes

total possible outcomes

C

C

11

21

12


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2 Find the probability of choosing a blue

marble from bag A.

Pr(B from A)number of favourable outcomes


C

C

51

81

58

=

=

=

3 Find the probability of choosing bag A and

then a blue marble.

Pr (A B) 1258

516

=

=

b 1 Find the probability of choosing bag A. b Pr(A )number of favourable outcomes


C

C

11

21

12

=

=

=

2 Find the probability of choosing a blue

marble from bag A.Pr(B from A ) =

number of favourable outcomes


CC

4

110

1

25

=

=

3 Find the probability of choosing bag Aandthen a blue marble.

Pr (A B) 1225

15

=

=

c A blue marble can be selected from bag

A orbag A.

c Pr(B) 51615

4180

= +

=

Recall from chapter 11 that, for conditional probability, A BA B

BPr ( )

Pr ( )

Pr ( )| =

, Pr (B) 0.

Rearranging this formula gives

Pr (AB) =Pr (A |B)Pr (B) [1]

From Worked example 30 above, notice that

Pr (B) =Pr (AB) +Pr (AB) or Pr (A) =Pr (AB) +Pr (AB ) [2]

Combining the information from equations [1] and [2], we have

Pr (A) =Pr (A|B)Pr (B) +Pr (A|B) Pr (B)

This expression is known as the Law of Total Probability and was briefly discussed in chapter 11.

Another way to visualise this rule is to use a tree diagram. The tree diagram below shows the situation

described in worked example 30. Notice how the probability of selecting a blue marble from bag A is

denoted as Pr (B |A). This is because the probability of selecting a blue marble from bag A is conditionalon selecting bag A to begin with.

Pr (A B) = =

A'

A12

12

12

610

516

410

58

58

Pr (A B') = =12

316

38

Pr (A'B') = =12

310

610

Pr (A'B) = = =12

15

210

410

38

(BA)

(B' A)

(B A')

(B' A')

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So, worked example 30 could also have been solved using a tree diagram or the Law of Total

Probability.

WORKED EXAMPLE 31

The probability that Suzanne will pass her examination given that she had help from her tutor

is11

15. The probability that Suzanne does not pass her exam given that she did not see her tutor is

2

5.

If the probability of Suzanne seeing her tutor is1

2

, what is the probability of her passing her exam?

THINK WR ITE/ DR AW

1 Define TandE. Write down all the information

that is given in the question.Let T=having help from the tutor.LetE=passing the exam.

E T E T TPr ( ) , Pr ( ) , Pr ( )111525

12

| = | = =

2 Draw a tree diagram to represent the information.

12

1115

415

12

35

2

5

T

T

ET

ET

ET

ET

3 Using the formula

Pr (E) =Pr (E|T ) Pr (T ) + Pr (E|T ) Pr (T ),substitute all the known values from the tree diagram.

EPr ( ) 111512

35

12

23

=

=

4 Interpret the result. The probability that Suzanne will pass her

exam is2

3.

Exercise 12H Applications to probability 1 WE26 Jenny, Hakan and Miriam are competing in a car race against 5 other drivers. Their friend Mary

predicts that they will cross the finish line first, second and third respectively. What is the probability

that Mary is right?

2 WE27 The letters of the word PRODUCE are randomly reordered. Calculate the probability that

the letters P and E will be together.

3 WE28 Six people selected from 5 men and 7 women are to form a committee. Work out the

probability that the committee will consist of 3 men and 3 women.

4 MC The letters A, B, C, D, E and F are randomly placed in a row. The probability that the letters A

and B will occupy the first and second positions respectively is:

A1

15B

1

3C

1

30D

1

6E

2

3

5 Six cards are randomly distributed from a standard pack of 52 playing cards. Determine the probability

that exactly one of the 6 cards is a queen. 6 From a toy set consisting of 4 dolls and 5 clowns, 2 toys are chosen at random. Find the probability that

the 2 toys are 2 clowns or 2 dolls.

7 MC From a group of 3 children and 8 adults, 5 will be chosen to receive prizes. The probability that

2 children and 3 adults will be awarded a prize is:

A

3

3

8

5

11

5

B

3

1

8

1

11

3

11

8

C

+

3

2

8

3

11

5

D

3

2

8

3

11

5

E

3

2

8

3

11

3

11

8

+

8 WE29 A group comprising 6 people is sitting around a table. Find the probability that two particular

people are sitting next to each other.

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Combinatorics

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9 Ten people are seated at a circular dining table. Find the probability that two particular people will

be sitting next to each other.

10 MC Six mothers and their 6 daughters randomly arrange themselves in a circle. The probability that

Susan is next to her daughter Jeanette is:

A1

4B

5

6C

2

3D

1

12E

2

11

11 Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one

letter E will occur in the selection of 4 letters.

12 A school captain and 2 vice-captains are to be chosen from a group of 5 boys and 6 girls. What is the

probability that all 3 positions will be taken by:a boys?b girls?c two boys and one girl?d at least two girls?

13 Four colours are randomly picked from the 7 different colours of the rainbow. Calculate the probability

that yellow will not be one of the colours chosen.

14 A dealer draws three cards from a deck of 52 cards. What is the probability that she draws:a no queens?b at least 2 queens?c exactly one heart?

15 Five letters are randomly selected from the letters of the word HOLIDAYS and placed in a row.Calculate the probability that the first letter chosen is a consonant.

16 MC Inside a box are nobjects of which mare white. If robjects are randomly taken out of the box

and placed in a row, the probability that the first object is white is:

An

mB

m

n!C

m

n

Dm n

n!

+

En m

n

17 MC A 5-digit number is randomly formed using the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9. If a digit cannot

be used more than once in the number:

a the probability that the number is even is:

A9

100B

57

195C

4

9D

35

78E

14

63

b the probability that the number is between 30 000 and 50 000 is:

A20

57B

19

73C

2

9D

35

78E

14

63

18 A debating team of 6 people is to be formed from a group consisting of 5 males and 6 females.

a What is the probability that the team will consist of at least one male?

b What is the probability that the team will have at least four females?

19 WE30 Two small crates (X and Y) contain apples (A) and bananas (B).

A crate is randomly selected, then from that crate a piece of fruit is

randomly selected. The table at right describes the distribution of fruit

between the crates.

a What is the probability of selecting crate X and from it, a banana?

b What is the probability of selecting crate Y and from it, a banana?

c What is the probability of selecting a banana?

d Find the probability of selecting a banana using Pr (B) =Pr (B|X)Pr (X) +Pr (B| Y)Pr (Y).20 Given Pr (B |A) = 3

5, Pr (B |A ) =1

3, and Pr (A) =3

4, find Pr (B) using the Law of Total Probability.

21 WE31 The probability that Tim is late for school is1

3, but he has an exam on Friday. The chance of

him passing his exam given that he is on time to school is5

7. If he is late, his chance of not passing the

exam is5

11. What is the chance that Tim will pass his exam?

22 Eleni loves chocolates. She particularly loves soft-centred chocolates. She is offered a box of

12 chocolates to select from, but all the chocolates are wrapped. The probability of selecting a soft-

centred chocolate given that it is dark chocolate is 25, and the probability of selecting a hard centre given

that it is milk chocolate is 47. If there are 7 milk chocolates in the box, find the probability of selecting a

soft-centred chocolate.

X Y

6 apples 4 apples

5 bananas 7 bananas

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23 Freds chance of being selected for the soccer team this season is 811

. The probability of Fred going on

the school trip given that he is selected for the soccer team is 712

, whereas the chance of him not

going on the school trip given that he is not selected for the soccer team is 34. What is the probability

that Fred will go on the school trip?

24 The chance of a sprinter winning a race given that his archrival runs is 13. If his archrival does not run,

the sprinter has a5

8chance of winning. His archrival is injured and has a

1

4chance of running at all.

Use the Law of Total Probability to find the probability that the sprinter wins the race.

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Summary

The addition andmultiplicationprinciples

The addition principle states that if two operations can be performed inAorBways respectively,

then both operations can be performed together inA+Bways. The multiplication principle states that if two operations can be performed inAandBways, then

both operations can be performed in succession inABways.

Permutations A permutation is the arrangement of objects in a definite order. The multiplication principle is

commonly used in calculating the number of possible permutations.

=

n

n r

P( )!

n

r

The number of ways of arranging nobjects that includepidentical objects of one type, qidentical

objects of another type, ridentical objects of yet another type and so on is:n

p q r! ! !

nobjects divided into mgroups, with each group having G1, G2, G3, . . . G mobjects respectively,

has m! G1! G2! G3! . . . Gm! arrangements.

Factorials The factorial of a positive whole number nis defined as:

n! =n(n1) (n2) (n3) . . . 3 2 1 with 0! =1 n! =n(n1)! = n(n1) (n2)! and so on.

Arrangements in acircle

ndistinguishable objects can be arranged in a circle in (n1)! ways. The same methods are applicable to arrangements in a circle as the methods used for

indistinguishable objects when there are restrictions on the possible arrangements.

Combinationsusing nC

r

The number of combinations when robjects are selected from nobjects is denoted

by nCrorn

r

. nCr=n

r

=n

r n r!( )! or

r

P

!

n

r

nCr=nCn r

Applications toprobability

The probability of an event: Pr (event) =

number of favourable outcomes

total number of possible outcomes

The Law of Total Probability states: Pr (A) =Pr (A|B)Pr (B) =Pr (A|B)Pr (B)

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Chapter review

1 There are 7 airlines that have flights from Australia to Singapore, 6 airlines that offer flights from

Singapore to Europe, and 5 airlines that service the route from Europe to America. Determine the

number of different travel arrangements possible to get from Australia to America via Singapore and

Europe.

2Seven people form a queue to board a bus. How many different queues are possible?

3 The digits 3, 5, 6 and 8 are used to form numbers greater than 100. If a digit may be used once only and

not all digits have to be used, how many different numbers can be formed?

4 Seven different books are to be placed on a shelf. If a particular book must occupy the first position,

find the number of permutations possible.

5 In how many ways can first, second and third prizes be awarded to 12 people competing in a marathon?

6 A team of at least 2 people must be chosen from a group of 5 mountaineers to mount a rescue

mission. How many different teams may be chosen?

MULTIPLECHOICE

1 Samantha can get to work by walking, by taking her car or by using public transport (train, tram, bus or

taxi). The number of different ways she can get to her work is:

A 3 B 5 C 4 D 6 E 2 2 Malcolm is guessing someones house number. He knows that the number is an odd number and is

between 30 and 60. Assuming that the same guess is not made twice, the maximum number of guesses

he can make is:A 15 B 20 C 30 D 45 E 25

3 The total number of 2-digit, 3-digit and 4-digit odd numbers that can be formed using the digits 6, 4, 5,

2, 1 when a digit cannot be used more than once is:

A 200 B 80 C 170 D 120 E 128

4 The value of998!

996!

is:

A undefined B 1000 999 C 996!

D 998 997 E 998 997 996 5 The value of 9! 7! is equivalent to:

A 71 7! B 2! C 7! 9 D 8! E 7 8!

6 The value of 7P5is:A 21 B 42 C 2520 D 1008 E 5040

7 The number of permutations using the letters of the word MISSISSIPPI is:

A 4! B 11! C11!

4!4!2!D 1!2!4!4! E

1

4!

8 Five letters are chosen from the letters of the word WATERING and placed in a row. The number of

ways in which this can be done if the last letter is to be W is:A 840 B 2520 C 210 D 40 E 625

9 A family consisting of a mother, father, 3 sons and 4 daughters lines up for a photograph. How many

ways can this be done if the daughters must be together?A 9! B 6!4! C 5!4!D 2!3!4! E 10!

10 Eleven members of a cricket team are to be seated in a circle. The number of possible arrangements is:A 5! B 10! C 11!

D11!

10!E

10!

11

11 The letters of the word MUSICAL are to be arranged in a circle. If the letters U and S must not be

together, the number of possible arrangements is:A 480 B 718 C 1440D 3600 E 5038

SHORTANSWER

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12 Joanna has decided to study at university. Her course requires that she undertake at least 2 subjects for

the year. If 4 subjects are being offered, the number of subject combinations is:A 36 B 24 C 15 D 11 E 20

13 Four pieces of fruit are selected from a box containing 5 oranges and 6 apples. The number of

selections that contain at least 2 oranges and 1 apple is:A 210 B 150 C 60 D 90 E 110

14 Five letters are randomly selected from the word ENERGISE. The probability that the letter E will

appear in the group of 5 letters is:

A5

8B

15

56C

3

8D

1

8E

55

56

1 A 3-, 4- or 5- digit number is to be formed using digits taken from 8, 4, 3, 6 and 7. If a digit may be

used more than once, how many different numbers can be made?

2 The 4 fastest runners in a race will qualify for the finals. If there are 11 competitors, determine the

number of different ways in which the race can finish.

3 Evaluate 9! +8! 6! +3 2!

4 Find the number of ways the letters of the word ARRANGEMENT can be placed in a row.

EXTENDEDRESPONSE

5 Anna, Belinda, Chien, Deanna and Erica are lining up for concert tickets. If Belinda and Deanna do not

want to be next to each other, what is the number of possible queues?

6 Ten children are arranging themselves in a circle. Calculate the number of ways this can be done if

three particular children are not to be next to each other.

7 Two students from a group of 8 students are to be class captain and vice-captain. From the remaining

candidates, two will become class monitors. Find the number of ways this can be done.

8 A class consists of 24 students. If an initial group of 4 must be chosen to go for a measles injection,

how many different combinations may be selected for that group?

9 A committee of 5 people is to be established using members from a group of 6 men and 7 women.

What is the probability that the committee will contain 2 men and 3 women?

10The letters of the word FEATURING are randomly rearranged. Find the probability that the letters ofthe word FEAT are together, though not necessarily in the order shown.

11 Two women and three men approach an ATM at the same time.

a How many different queues are possible if the position of each person in the queue is taken into

account?

b How many queues of at least two people are possible if the position of each person in the queue is

not taken into account?

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12 From a group of 20 female students, 2 female staff, 18 male students and 3 male staff, a committee of

6 is to be formed.

a Find the number of different committees if:

i there are no restrictions

ii all committee members must be students

iii one female and one male staff member must be on the committee iv there is an equal number of males and females on the committee

v one particular student must be on the committee

vi one particular student must not be on the committee

vii the committee must comprise 2 male staff members, 2 male students, 1 female staff member

and 1 female student.

b Find the probability that:

i only students are selected for the committee

ii all the staff are selected for the committee

iii exactly 2 staff and 4 students are selected.

13 In the game of Tattslotto, a barrel contains forty-five balls numbered 1 to 45, of which eight are

randomly drawn. The first six of these numbered balls are the winning numbers. The final two drawn

are called supplementary numbers. When you purchase a standard ticket, you may select six numbersin each game. Prizes are awarded according to how many of your six numbers match those drawn from

the barrel.

To win the first prize (division one), all six of your numbers must match the six winning numbers

drawn from the barrel.

To win the second prize (division two), five of your numbers must match the winning numbers and

your remaining number must match one of the supplementary numbers.

To win the third prize (division three), five of your numbers must match the winning numbers.

(Your remaining number does not match any of the numbers drawn.)

a What is the probability of winning division one?

b What is the probability of winning division two?

c What is the probability of winning division three?

d What is the probability of winning at least a division three prize?

DIGITAL DOCdoc-9820

Test YourselfChapter 12

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ICT activities

Chapter opener

DIGITAL DOC

10 Quick Questions doc-9813:Warm-up with ten quick questions on

combinatorics (page 529)

12A The addition and multiplication principlesTUTORIAL

WE4 eles-1454:Use the multiplication and addition rules to

calculate the number of different lunches and the number of different

dishes that can be ordered at a cafeteria (page 531)

12B Permutations

TUTORIAL

WE7 eles-1455:Use permutations to determine the number of

ways three awards and two prizes can be distributed to five different

people (page 534)

12C Factorials

DIGITAL DOC WorkSHEET 12.1 doc-9814:Determine the number of combinations

in different scenarios and calculate expressions involving factorials

(page 537)

12D Permutations using nPr

TUTORIAL

WE14 eles-1456:Use permutations to determine the number

of different way five positions can be determined from ten

people (page 539)

DIGITAL DOCS

SkillSHEET 12.1 doc-9815:Practise calculating nPr(page 540).

doc-9816:Investigate combinatorics using a spreadsheet(page 540)

12E Permutations involving restrictions

INTERACTIVITY

Permutations involving restrictions int-0271:Consolidate

your understanding of permutations involving restrictions

(page 541)

TUTORIAL

WE18 eles-1457:Determine the number of ways the letters in

a particular word can be arranged if two specific letters cannot be

adjacent (page 543)

12FArrangements in a circleTUTORIAL

WE21 eles-1458:Determine the number of ways six people can

be arranged around a table, if two specific people must be seated

next to each other (page 546)

DIGITAL DOC

WorkSHEET 12.2 doc-9817:Calculate permutations and evaluate

expressions involving nPr(page 547)

12G Combinatorics using nCrTUTORIAL

WE25 eles-1459:Calculate the number of different committees

that can be formed from a group of five men and four women, given

three varying constraints (page 549)

DIGITAL DOCS doc-9816:Investigate combinatorics using a spreadsheet (page 549)

SkillSHEET 12.2 doc-9818:Practise identifying and listing possible

outcomes (page 550)

Investigation doc-9819:Investigate Pascals triangle (page 550)

12H Applications to probability

TUTORIAL

WE30 eles-1460:Apply the law of total probabilities and the

probability of an event to calculate probabilities of selecting specific

coloured marbles from two bags (page 552)

DIGITAL DOC

doc-9816:Investigate combinatorics using a spreadsheet (page 554)

Chapter review

DIGITAL DOC

Test Yourself doc-9820:Take the end-of-chapter test to test your

progress (page 560)

To access eBookPLUS activities, log on to www.jacplus.com.au

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Answers CHAPTER 12

COMBINATORICS Exercise 12A The addi tion andmultiplication principles

1 21 2 112 3 97 4 E 5 5 6 B

7 150 8 72 9 a 24 b 144010 E11 D12 aE is English,

M is Mathematics,L is Language,S is Science.E, M, L, S, EM, EL, ES, ML, MS, LS,EML, EMS, ELS, MLS, EMLS

b There are 15 ways in total.13 a Walk/walk, walk/bus, train/walk, train/

bus, bus/walk, bus/bus b

WalkBus

Walk

Walk

Bus

WalkBus

Train

Bus

14 16 15 30

Exercise 12B Permutations

1 24 2 720 3 336 4 100 000 5 120 6 40 320 7 a 64 b 56 8 320 9 a 750 b 180

10 80 11 320 12 32013 C 14 C 15 C16 256

17 Lose $17 144.0018 a i 479 001 600 ii 362 880 b 184 02319 a 60 000 b In the 15th year

Exercise 12C Factorials 1 a 24 b 362 880 c 479 001 600 d 8 e 96 f 4318 g 567 2 a 0 b 156 c 2880 d 0 e 40 206 f 2 177 298 g 35 376 h 439 085 448 3 B

4 D 5 a 12 b 5 c 840 d 120 e 4 6 a10 302 b 999 000 c 4500 d 15 438 000 e 61 629 480 f 25 000 7 a 15 b 756 c 2970 d 999 000 e 3906 8 E 9 a 214 b 104 c 1596 d 491 952 e 122

10 a 40 b 90 c 41 d 1681 e 4

Exercise 12D Permutations using nPr

1 a 360 b 56c 504 d 24

e 6 375 600 f 6 g 17 h 19 950 i 0 j 36 k 94 109 400 l 7 880 400 2 303 600 3 20 4 300 5 a 24 b 36 6 E 7 13 366 080 8 151 200 9 672010 3 628 80011 a362 880 b 720

Exercise 12E Permutations involving

restrictions 1 126 2 4620 3 B 4 5040 5 60 6 a 2 793 510 720 b 147 026 880 7 48 836 9 480 10 16811 C 12 6 289 92013 78 624 14 A

Exercise 12F Arrangements in a circle 1 120 2 E 3 3360 4 6930 5 4320 6 252 7 5445 8 D 9 1260 10 D

114

165

Exercise 12G Combinations using nCr

1 a 10 b 4 c 6 d 1 e 1 2 a 15 b 21 c 45 d 84 e 924 3 a 30 b 1485 c 41 664 d 73 815 e 118 755 4 A 5 a3, 3 b 4, 4 c 10, 10 d 84, 84 6 a 20C7=20C13 b 100C9=100C91 7 792 8 15 9 21 10 593 75511 60 12 52513 22 050 14 490 77615 a 252 b 12016 10 17 718 a 1001 b 420 c 595 d 7019 a 28 b 5620 a 142 506 b 128%21 E 22 7023 A

Exercise 12H Applications to probability

11

336 2

2

7

325

66 4 C

5 0.336 64

9

7 D 82

5

92

910 E

112

33

12 a2

33 b

4

33

c4

11 d

19

33

133

7

14 a 0.783 b 0.013 c 0.436

155

8

16 C

17 a C b C

18 a461

462 b

181

462

19 a5

22 b

7

22

c6

11 d

6

11

208

1521

152

231

225

1223

65

132

2453

96

CHAPTER REVIEWSHORT ANSWER

1 210 27! =5040 3 48 46! =720 5 1320 626

MULTIPLE CHOICE

1 D 2 A 3 B 4 D 5 A 6 C 7 C 8 A 9 B 10 B11 A 12 D13 A 14 E

EXTENDED RESPONSE 1 3875 2 7920 3 402 486 4 2 494 800 5 72 6 332 640 7 840 8 10 626

9 0.408 or 175

429

10 0.0476 or 1

21

11 a 120 b 26

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12 a i 6 096 454 ii 2 760 681 iii 442 890 iv 2 048 200 v 850 668 vi 5 245 786 vii 18 360 b i 0.453 ii 0 (negligible) iii 0.121

13 a 1.23 10 7

b 1.47 10 6

c 2.73 10 5

d 2 .8 9 1 0 5

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Exam practice 5 565

Exam practice 5 CHAPTERS 112

1 AandBare two events such that Pr (A) =0.3, Pr (B) =0.3 and Pr (AB)=0.6. Determine the value ofPr(A B). 2 marks

2 Fifty-five Year 11 students at Grampians Rise Secondary College attended the study camp. On the

camp, students were able to select up to three physical activities: bike riding, rock climbing or

hiking.

29 students selected hiking.

22 students selected rock climbing.

28 students selected bike riding.

7 students selected both bike riding and rock climbing.

8 students selected only bike riding and hiking.

5 students selected all three activities.

Some of this information is represented on the Venn diagram at right.

Hrepresents hiking,Rrepresents rock climbing andBrepresents bike riding.

a Determine the values of a, b, cand d. 3 marks

b Determine the number of students who did not select any of the activities. 1 mark

c Determine the probability that a student selected at random selected bike riding and hiking. 1 mark

3 65% of learner drivers have more than 300 hours driving practice.

40% of minor car accidents are caused by learner drivers.

50% of lear

mm11_chap12

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