A permutation is an ordered arrangement of items where order matters and no item is used more than once. The number of permutations of n items taken r at a time is calculated as nPr = n!/(n-r)!. Factorial notation, written as n!, represents the product of all positive integers down to 1. For example, 5! = 5 x 4 x 3 x 2 x 1. Permutations with duplicate items use the formula n!/(p!q!r!...) where p,q,r... represent the numbers of identical items.

1. 11.2 Permutations

2. What exactly is permutation? A permutation is an ordered arrangement of items that occurs when no item is used more than once and the order of arrangement makes a difference.

3. When do we use it? Example 1: Suppose you are in charge of planning a school event where a group of freshmen, sophomores, juniors and seniors will present a song and dance. How many ways can you put together this song and dance event? Solution: 4 x 3 x 2 x 1 = 24 ways

4. Another Example Suppose you want to arrange the 7 Harry Potter books where the order makes a difference. How many ways can you arrange these? Answer: 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 ways

5. One more example Going back to example 2, suppose that you want the first book (The Sorcerer’s Stone) to be the first book on the arrangement, how many possible ways can you arrange the book? Answer: 720 ways

6. Try to do these problems on your own Suppose that Mariah Carey, Whitney Houston, Mary J. Blige, Beyonce and Rihanna will appear in concert at Seton. How many ways can you put together this event? Suppose that Beyonce prefers to be the first performer and Whitney as the last, how many ways can you put together the concert?

7. Answer: 1. 120 2. 6

8. Factorial Notation Instead of multiplying 5 x 4 x 3 x 2 x 1, is there an easy way to solve this problem? Of course! Try using factorial notation. 5 x 4 x 3 x 2 x 1 = 5 !

9. Definition of Factorial Notation If n is a positive integer, the notation n! is the product of all positive integers down through 1 n! = n (n-1) (n-2)…1 By definition, 0!=1

10. Evaluating Factorial without a calculator 1. 10! 2. 4! 3. 498! 8! 6! 497! Answer: 1) 90 2) 1/30 3) 498

11. Using the Calculator to solve factorials Enter the number. Press the MATH button. Go to PRB Press 4:! Press Enter

12. Permutation of n Things taken r at a time You and 9 of your friends have decided to form a new club at Seton. The group needs 3 officers: President, Secretary and Treasurer. In how many ways can these offices be filled? Answer: 10 x 9 x 8 = 720

13. Another way to solve the problem of permutations of n things taken r at a time The number of possible permutations if r items are taken from n items is nPr = n! (n- r)!

14. Another example Suppose you are asked to list, in order of preference, the three best songs you have downloaded this month. If you downloaded 30 songs, how many ways can the three best be chosen and ranked? Solution: 30 nPr 3 = 24, 360 To find 30 nPr 3, press 30 (the n), then press MATH, go to PRB, press 2:nPr, press 3 (the r) and press enter.

15. Permutation of Duplicate Items The number of permutations of n items, where p items are identical, q items are identical, r items are identical and so on, is given by n!__ p! q! r!...

16. Example 6 from text p. 571 In how many distinct ways can the letters of the word MISSISSIPPI be arranged? Solution: 11! 4! 4! 2! = 34, 650

17. Assignments Class work: Checkpoints 1-6, pages 566 – 571 HW: p. 571, #s 1-11 (all) p. 572, #s 13-39 (odd); 41-54 (all) Quiz 1 (11.1 and 11.2) tomorrow