12.1 & 12.2: The Fundamental Counting Principal, Permutations, & Combinations.

12.1 & 12.2: The Fundamental Counting Principal, Permutations,

& Combinations

12.1 & 12.2 Vocabulary

• Outcome: The result of a single trial.• Sample space: The set of all possible outcomes.• Event: Consists of one or more outcomes of a trial.• Independent events: Is not affected by any outcome.• Dependent events: The outcome of one event affects the outcome

of another event.• Permutation: A group of objects or people are arranged in a certain

order.• Linear Permutation: A group of objects or people are arranged in a

line.• Combination: An arrangement or selection of objects in which

order DOES NOT matter.

The Fundamental Counting Principal

• If you have 2 events: 1 event can occur m ways and another event can occur n ways, then the number of ways that both can occur is m*n

• Event 1 = 3 types of meats• Event 2 = 2 types of bread

• How many diff types of sandwiches can you make?

• 3(types of meat)*2(types of bread) = 6

3 or more events:

• 3 events can occur m, n, & p ways, then the number of ways all three can occur is m*n*p

• 3 meats• 2 breads• 3 cheeses• How many different sandwiches can you

make?• 3*3*2 = 18 sandwiches

• At a restaurant at Six Flags, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different deserts.

• How many different dinners (one choice of each) can you choose?

• 8*2*12*6 =1152 different dinners

Example

Fund. Counting Principal without Repetition

• Ohio Licenses plates have 3 #’s followed by 3 letters.

• 1. How many different licenses plates are possible if digits and letters can be repeated?

• There are 10 choices for digits and 26 choices for letters.

• 10*10*10*26*26*26= 17,576,000 different plates

How many plates are possible if digits and numbers cannot be repeated? Are the events independent or dependent?

• There are still 10 choices for the 1st digit but only 9 choices for the 2nd, and 8 for the 3rd.

• For the letters, there are 26 for the first, but only 25 for the 2nd and 24 for the 3rd.

• 10*9*8*26*25*24= 11,232,000 plates

Permutations

• Orders arrangement of items where the order is important.– You can use the Fund. Counting Principal to

determine the number of permutations of n objects.

• How many ways can we arrange ABC?

Example

• ABC• ACB• BAC• BCA• CAB• CBA

There are 3 choices for 1st #2 choices for 2nd #1 choice for 3rd.3*2*1 = 6 ways to arrange the letters

Try on your own.

• How many different ways can 11 futbol players be arranged in a lineup?

• 11! = 11*10*9*8*7*6*5*4*3*2*1

= 39,916,800 different ways

Factorial with a calculator:

•Hit math then over, over, over.

•Option 4

Factorials

•=•=

Using Permutations

• An ordering of n distinct objects taken r at a time is a permutation of the objects.

• P(n,r)= • nPr=

Olympic skiing competition.

• How many different ways can 12 skiers finish 1st, 2nd, & 3rd (gold, silver, bronze)

• Any of the 12 skiers can finish 1st, then any of the remaining 11 can finish 2nd, and any of the remaining 10 can finish 3rd.

• So the number of ways the skiers can win the medals is

• 12*11*10 = 1320

Back to the last problem with the skiers

• It can be set up as the number of permutations of 12 objects taken 3 at a time.

•12P3 = 12! = 12! =

(12-3)! 9!•

12*11*10*9*8*7*6*5*4*3*2*1 =

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

•12*11*10 = 1320

10 colleges, you want to visit.

• How many ways can you visit

6 of them:• Permutation of 10 objects taken 6 at a

time:•

10P6 = 10!/(10-6)! = 10!/4! =

• 3,628,800/24 = 151,200

How many ways can you visitall 10 of them:

•10P10 =

• 10!/(10-10)! = • 10!/0!=• 10! = ( 0! By definition = 1)• 3,628,800

Permutations with Repetition

• The number of DISTINGUISHABLE permutations of n objects where one object is repeated p times, another is repeated q times, and so on :

• n! p! * q! * …

So far in our problems, we have used distinct objects.

• If some of the objects are repeated, then some of the permutations are not distinguishable.

• There are 6 ways to order the letters M,O,M

• MOM, OMM, MMO• MOM, OMM, MMO• Only 3 are distinguishable. 3!/2! = 6/2 = 3

Find the number of distinguishable permutations of the letters:

• How many different ways can the word Mississippi be arranged.

• MISSISSIPPI : 11 letters with I repeated 4 times, S repeated 4 times, P repeated 2 times

• 11! = 39,916,800 = 34,650• 4!*4!*2! 24*24*2

Find the number of distinguishable permutations of the letters:

• SUMMER :

• 360

• WATERFALL :

• 90,720

Combinations

• An arrangement of objects in which order is NOT IMPORTANT

Whats the Difference? "My fruit salad is a combination of apples,

grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2. The combo is really a permutation lock

Combination with Multiple Events

• Five cards are drawn from a standard deck (52 cards). How many hands consist of three clubs and two diamonds?

• (Combination of event 1) x (Combination of event 2)

• and • x= 22,308 hands

How lotteries work.

• The numbers are drawn one at a time, and if you have the lucky numbers (no matter what order) you win!

• So what is your chance of winning?

The easiest way to explain it is to:

• assume that the order does matter (ie permutations),

• then alter it so the order does not matter.