Thinking Mathematically

Permutations

Permutations

• No item is used more than once.

• The order of arrangement makes a difference.

Example Counting Permutations

Based on their long-standing contribution to rock music, you decide that the Rolling Stones should be the last group to perform at the four-group Offspring, Pink Floyd, Sublime, Rolling Stones concert. Given this decision, in how many ways can you put together the concert?

Solution

We use the Fundamental Counting Principle to find the number of ways you can put together the concert. Multiply the choices:

3 2 1 1 = 6

Thus, there are six different ways to arrange the concert if the Rolling Stones are the final group to perform.

Example Counting Permutations

You need to arrange seven of your favorite books along a small shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference to you?

Solution

You may choose any of the seven books for the first position on the shelf. This leaves six choices for second position. After the first two positions are filled, there are five books to choose from for the third position, four choices left for the fourth position, three choices left for the fifth position, then two choices for the sixth position, and only one choice left for the last position.

7 6 5 4 3 2 1 = 5040There are 5040 different possible permutations.

Factorial Notation

If n is a positive integer, the notation n! is the product of all positive integers from n down through 1.

n! = n(n-1)(n-2)…(3)(2)(1)

0!, by definition is 1.

0!=1

Example Using Factorial Notation

Evaluate the following factorial expressions without using the factorial key on your calculator.

26!21!

=26 • 25• 24 • 23 • 22 • (21!)(21!)

=26 • 25• 24 • 23 • 22 =7,893,600

Permutations of n Things Taken r at a Time

The number of permutations possible if r items are taken from n items is:

Example Using the Formula for Permutations

You and 19 of your friends have decided to form an Internet marketing consulting firm. The group needs to choose three officers - a CEO, an operating manager, and a treasurer. In how many ways can those offices be filled?

SolutionYour group is choosing r = 3 officers from a group of n = 20 people (you and 19 friends). The order in which the officers are chosen matters because the CEO, the operating manager, and the treasurer each have different responsibilities. Thus we are looking for the number of permutations of 20 things taken 3 at a time.

20!

(20 −3)!=

20!17!

=20 • 19 • 18 • 17!

17!=20 • 19 • 18 =6840

Example Using the Formula for Permutations

You are working for The Sitcom Television Network. Your assignment is to help set up the television schedule for Monday evenings between 7 and 10P.M. You need to schedule a show in each of six 30-minute time blocks. You can select from among the following situation comedies: “Home Improvement,” “Seinfeld,” “Mad About You,” “Cheers,” “Friends,” “Frasier,” “All in the Family,” “I Love Lucy,” “M*A*S*H,” “The Larry Sanders Show,” “The Jeffersons,” “Married with Children,” and “Happy Days.” How many different programming schedules can be arranged?

Solution Using the Formula for Permutations

You are choosing r = 6 situation comedies from a collection of n=13 classic sitcoms. The order in which the programs are aired matters. Family-oriented comedies have higher ratings when aired in earlier time blocks, such as 7 to 7:30. By contrast, comedies with adult themes do better in the later time blocks.

13P6 = 13!(13−6)!

=13!7!

=13 • 12 • 11• 10 • 9 • 8• 7!7!

=13 • 12 • 11• 10 • 9 • 8 =1,235,520

Thinking Mathematically

Permutations