EXAMPLE 1 Counting Permutations

Music

You have five CDs. You can use the counting principle to count the number of permutations of 5 CDs. This is the number of different orders in which you can listen to the CDs.

Choices for 1st CD

Choices for 2nd CD

Choices for 3rd CD

Choices for 4th CD

Choices for 5th CD

5 4 3 2 1 = 120

ANSWER

You can listen to the CDs in 120 different orders.

EXAMPLE 2 Counting Permutations

Band Competition

Twelve marching bands are entered in a competition. You can use the counting principle to count how many ways first, second, and third places can be awarded.

Choices for 1st place

Choices for 2nd

place

Choices for 3rd place

12 11 10 = 1320 Counting principle

ANSWER

There are 1320 ways to award the three places.

GUIDED PRACTICE for Examples 1 and 2

1. Evaluate 3!.

3! = 3 2 1 = 6

2. Evaluate 6!.

6! = 6 5 4 3 2 1 = 720

3. Evaluate 1!.

1! = 1

GUIDED PRACTICE for Examples 1 and 2

4. Chris wants to see four movies that were released in the past month. In how many different orders can Chris watch the movies?

Choices for 1st place

Choices for 2nd

place

Choices for 3rd place

Choices for 4th place

4 3 2 1 = 24 Counting principle

Chris can watch the movies in 24 different orders.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

5. In how many ways can 7 runners finish in first, second, and third place?

Choices for first

Choices for second

Choices for third

7 6 5 = 210 Counting principle

ANSWER

There are 210 ways to finish the first, second, and third place.