PERMUTATIONSPERMUTATIONS

The Multiplication Principles states that:

If an operation can be carried out in r ways, a second operation can be carried out in s ways, then there are (r s) ways of carrying out the first operation followed by the second operation.

In how many ways can the three letters A, B, and C be arranged in a line?

In how many ways can the three letters A, B, and C be arranged in a line?

By listing:By listing:

ABCABC BACBAC CABCAB

ACBACB BCABCA CBACBA

By using a tree diagram:By using a tree diagram:

B

A

C

B

C

A

CA

B

C

B

C

A

B

A

ABC

ACB

BAC

BCA

CAB

CBA

By the multiplication principle:By the multiplication principle:

The first letter can be chosen in The first letter can be chosen in three ways.three ways.

The second letter can be chosen in The second letter can be chosen in two ways.two ways.

The third letter can be chosen in The third letter can be chosen in only one way.only one way.

Therefore, the number of ways of Therefore, the number of ways of arranging the three letters is (3 arranging the three letters is (3 2 2 1) ways, that is, 6 ways.1) ways, that is, 6 ways.

The different arrangements The different arrangements of the objects are called of the objects are called permutationspermutations..

Rule 1

The number of ways of arranging n unlike objects in a line is n!

n! = n (n – 1) (n – 2) … 3 2 1Note: 0! = 1

Example 1:

A witness reported that a car seen speeding away from the scene of the crime had a number plate that began with three letters B, W and P followed by four digits 2, 5, 3 and 9. He could not however remember the order of the letters or the digits. How many cars would need to be checked to be sure of including the suspect’s car?

Rule 2

The number of ways of arranging in a line n objects, of which p are alike, is

p!

n!

The number of ways of arranging in a line n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, and so on, is

...r!q!p!

n!

There are 3! ways of arranging the three letters A, B and C.

If B and C are replaced by X, we have

ways of different arrangement.2!

3!

AXX XXA XAX

2 Xs

Example 2:

a)In how many ways can the letters of the word STATISTICS be arranged?

b)If the letters of the word MINIMUM are arranged in a line at random, how many of the arrangements begin with MMM?

Example 3:

In how many ways can ten pupils be placed at random in a line such that the two youngest pupils are separated?

Example 4:

If a four-digit number is formed from the digits 1, 2, 3 and 5 and repetitions are not allowed, find the number of ways of arranging the four digits such that the number formed is divisible by 5.

Example 5:

A group of 12 people consisting of 6 married couples is arranged at random in a line for a photograph. Find the number of different arrangements that each wife is standing next to her husband.

Example 6:

A typist has five letters and five addressed envelopes. In how many different ways can the letters be placed in each envelope without getting every letter in the right envelope?

Rule 3

The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!

If four people A, B, C and D are to be seated at a round table, the following four arrangements are the same.

A

B

C

D A

B

C

D

A

B

C

D A

B

C

D

Rule 4

The number of ways of arranging n unlike objects in a ring, when clockwise and anticlockwise arrangements are the same, is

2

1)!(n

If A, B, C and D are four different coloured beads which are threaded on a ring, then the following two arrangements are the same, since one is the other viewed from the other side.

A

B

C

D D

C

B

A

Example 7:

Six bulbs are planted in a ring and two do not grow. In how many ways can the bulbs be arranged such that the two that do not grow are next to each other?

Example 8:

One white, one blue, one red and two yellow beads are threaded on a ring to make a bracelet. Find the number of arrangements that the red and white beads are next to each other.

W W

W

WW

W

R R

R

RRR

B

B B

B

B

B

Y

Y

Y

Y Y

Y

Y

YY

Y

Y

Y

Example 9:

Nine children play a party game and hold hands in a circle. In how many ways can they be arranged such that Mary will be holding hands with her friends Natalie and Sarah?

In general, the number of different permutations of n distinct objects in a line is written as nPn where

nPn = n!

The number of different permutations of r objects taken from n distinct objects is written as nPr where

rnP

r)!(n

n!Pr

n

Suppose now that you want to make a four-letter word from 7 letters A, B, C, D, E, F, and G.

7 choices

The number of permutations of 4 letters chosen from 7 letters is therefore 7 6 5 4 = 840.

6 choices

5 choices

4 choices

4)!(7

7!

3!

7!

123

12345674567

4)!(7

7!P4

7

Now,

That is, the number of permutations of 4 letters chosen from 7 letters is

Example 10:

Ten athletes are running in a 100 metre race. In how many different ways can the first three places be filled?

Example 11:

Given six cards as follow:

Find the number of different arrangements of three out of six cards in a row.

1 2 2 3 4 5

Case 1 (0 or 1 digit ‘2’ is taken):

number of arrangements = 5P3 = 60

Case 2 (2 digit ‘2’ are taken):

number of arrangements

122!

PP 3

3

14

Therefore, the total number of arrangements = 60 + 12 = 72.