1. Permutations with Repetitions

2. 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

3. 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.

4. Permutations with Repetitions Example 2: Along how many different routes can one walk a total of 9 blocks by going 4 blocks north 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.

5. 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.

6. 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.

7. 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:

8. 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 2 nd