3. permutation and combination

PERMUTATION &

COMBINATION

MULTIPLICATION RULE

If an operation can be performed in r different ways and a second operation can be performed in s different ways, then both the operations can be performed successively in r x s ways.

Multiplication Rule/Principle

10

23

7

9

2 3 4 24=

The number of ways the outfit can be chosen =

PERMUTATION

In permutation, the order of the objects or outcomes is important. Each different order represents a different outcome.

Permutation – variation, order

Arrangement

PERMUTATION

Permutation – the arrangement is important

How many ways can the letter X and Y be arranged?

There are two ways two different permutation

COMBINATIONS

In Combinations, we do not arrange the selections in order.

Combination – grouping,selection

Choices

Combination Arrangement is not important

Or

Are the same one combination

Tom & Jerry Jerry & Tom

Tom & Jerry

Jerry & Tom

They are the same

cat

& the mouse

OR

DIFFERENCES BETWEEN PERMUTATIONS AND COMBINATIONS

PERMUTATIONS COMBINATIONS

Arranging people, digits, numbers, alphabets, letters, colours.

Keywords: Arrangements, arrange,…

Selection of menu, food, clothes, subjects, teams.

Keywords:Select, choice,…

Permutation

Number of ways to arrange n different objects

Number of ways to arrange 3 different objects

6 ways

Number of ways to arrange 4 different objects

24

Number Number of of objects ways 1 ……… ………………………

2 ………. …….…………………

3 ……….. ……………….………

4 ……….. ……………….………

1

2 x 1

1

2

6 3 x 2

2 x 1

24 4 x 6

3 x 2

x 1

5 ……….. ……………………….

120 5 x 24

4 x 3 x 2 x 1

6 x 5 x 4 x 3 x 2 x 1 6 ……….. ……………………….

= 6!

= 5!

= 4!

= 3!

= 2!

= 1!

Factorial

The number of ways to arrange n objects = n !

1. To arrange 10 different objects = 10 !

2. To arrange digits 2, 5, 6, 8

= 4 !4 different

objects3. To arrange 12 finalists

12 different objects

= 12 !

DNADNA

23 pairs of chromosomes23 pairs of chromosomes

23! different ways to arrange23! different ways to arrange

The number of ways to arrange 23 different objects ?

25852016738884976640000

23 !

= 2.6 x 1022

Permutation

Number of ways to arrange r objects from n objects

8choices

7choices

6choices

5choices

4choices

Number of ways to arrange 5 students from 8 students.

8 x 7 x 6 x 5 x 4 = 8 x 7 x 6 x 5 x 4 = 6720

8choices

7choices

6choices

5choices

4choices

number of ways to arrange r from n objects

n (n-1) (n-2) (n-3) (n-(r-1))

)!(

!

rn

nPr

n

1. Questions related to Forming Numbers with digits and conditions

Use Multiplication Rules

Condition 1Find the number of ways to form 5 letter word from the letters W, O,R, L, D, C, U, P with the condition that it must starts with a vowels.

is filled first

W R L D C P

O

O

U 22 7 6 5 4

U =168

0

Find the number of ways to form 6 letter word from the letters B, E, C, K, H, A, M with the condition that it must starts with a consonant.

B CKHM

5

EA

6 5 4 3 2

5x6x5x4x3x2 = 1200

2. Questions related to Forming Numbers with digits and conditions

Use Multiplication Rules Conditions :

Sit side by side, next to each other – group together and consider as 1 object for arranging with other objects, make sure remember the arrangement of the grouped objects itself.

6 !

2!2! = 1440

4. To arrange PENALTY such that vowels are side by side

1 2 3 4 5 6

3. Complimentary Methods

Use:

The number of arrangements of event A

= Total arrangements – arrangement of A’

A

A’

S

Example

Find the number of the arrangement of all nine letters of word SELECTION in which the two letters E are not next to each other.

Solutions:

Total no. of arrangements – No. of arrangements with two E next to each other

141120

!82

!9

Combinationsn objects choose n = 1

N = 4

Choose 1:

A A B B C C DD

AA BB CC DD

Choose 2: AABB AACC AADD BBCC

Choose 3: AABBCC AACCDD BBCCDD

BBDDCCDD

AABBDD

Choose 4: AABBCCDD

= 4 = = 4 = 44CC11

= 6 = 6 = = 44CC22

= 4 = 4 = = 44CC33 = 1 = 1 = = 44CC44

Combinations

)!(!

!

rnr

nCr

n

Conditional Combination 1A football team has 17 local players and 3 imported players. Eleven main players are to be chosen with the condition that it must consist of 2 imported players. Find the number of ways the main player can be chosen.

import local

n

r

3 172 9

3C217C9

= 72930

Condition Combination 2A committee consisting of 6 members is to be chosen from 3 men and 4 women. Find the number of ways at least 3 women are chosen.

W3 M3, or W4 M2,

4C3X 3C3 + 4C4 X 3C2

= 7

CONCLUSIONS

1. Compare and Contrast between Permutations and Combinations.

DIFFERENCES BETWEEN PERMUTATIONS AND COMBINATIONS

PERMUTATIONS COMBINATIONS

1. Order is importent

2. Arranging people, digits, numbers, alphabets, letters, colours, …

3. Keywords: Arrangements, arrange,…

1. Order is not important.

2. Selection of menu, food, clothes, subjects, teams, …

Keywords:Select, choice,…

2. Formula

Difference between the two formulae:

Use the calculator to find the values of permutations and combinations.

)!(!

!

rnr

nCr

n

)!(

!

rn

nPr

n

3. If not sure, try to use the Multiplication Rules

Know the ways how to handle conditions like:

Sit side by side, next to each other, even/odd numbers, more/less than, starts/ends with vowel/consonants, …

Find the number of ways to form 6 letter word from the letters B, E, C, K, H, A, M with the condition that it must starts with a consonant.

B CKHM

5

EA

6 5 4 3 2

5x6x5x4x3x2 = 1200

4. For complicated cases:

Simplify by using Complimentary Methods

The number of arrangements of event A

= Total arrangements – arrangement of A’

A

A’

S

PROBABILITY