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