11.2 permutations
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permutationsTRANSCRIPT
11.2 Permutations
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.
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
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
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
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?
Answer:
1. 120
2. 6
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 !
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
Evaluating Factorial without a calculator
1. 10! 2. 4! 3. 498! 8! 6! 497!
Answer: 1) 90 2) 1/30 3) 498
Using the Calculator to solve factorials
Enter the number.Press the MATH button.Go to PRBPress 4:!Press Enter
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
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)!
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, 360To find 30 nPr 3, press 30 (the n), then press
MATH, go to PRB, press 2:nPr, press 3 (the r) and press enter.
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!...
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
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