11.2 Permutations

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