Permutations with Repetitions

Permutations with Repetitions

Permutation Formula

The number of permutations of “n” objects, “r” of

which are alike, “s” of which are alike, ‘t” of which

are alike, and so on, is given by the expression

!

! ! ! ...

n

r s t

Permutations with Repetitions

Example 1: In how many ways can all of the

letters in the word SASKATOON be arranged?

Solution: If all 9 letters were different, we could

arrange then in 9! Ways, but because there are 2

identical S’s, 2 identical A’s, and 2 identical O’s,

we can arrange the letters in:

Therefore, there are 45 360 different ways the letters can be arranged.

! 9!45360

! ! ! ... 2! 2! 2!

n

r s t

Permutations with Repetitions

Example 2: Along how many different routes canone walk a total of 9 blocks by going 4 blocksnorth and 5 blocks east?

Solution: If you record the letter of the direction in

which you walk, then one possible path would be

represented by the arrangement NNEEENENE. The

question then becomes one to determine the number of

arrangements of 9 letters, 4 are N’s and 5 are E’s.

Therefore, there are 126 different routes.

9!126

5! 4!

Circular and Ring Permutations

Circular Permutations Principle“n” different objects can be arranged in

circle in (n – 1)! ways.

Ring Permutations Principle

“n” different objects can arranged on a

circular ring in ways.( 1)!

2

n

Circular and Ring Permutations

Example 1: In how many different ways can

12 football players be arranged in a circular

huddle?

Solution: Using the circular permutations principle there are:

(12 – 1)! = 11! = 39 916 800 arrangements

If the quarterback is used as a point of reference, then the other 11 players can be arranged in 11! ways.

Circular and Ring Permutations

Example 2: In how many ways can 8 different charms be arranged on a circular bracelet?

Solution: Using the ring permutation principle there are:

( 1)! (8 1)! 7!2520

2 2 2

nways

Homework

Do # 1, 2, 4, and 6 – 8 on page 199 from Section 6.3 and # 1 – 7 on page 204 from Section 6.4 for Tuesday June 2nd