Warm Up

1. How many 2-side-dish meals can be made from 6 choices of side dishes?

2. Kim has shorts in blue, black, and tan. She has shirts in blue, yellow, red, and green. How many different combinations can she make? 3. If you go to the movies and are allowed to get 2 snacks and there are 9 snacks to choose from, how many combinations are there to pick from?

Course 2

10-7 Permutations

Learn to find the number of possible permutations and the probability that a specific permutation will occur.

Course 2

10-7 Permutations

Vocabulary

permutationfactorial

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Course 2

10-7 Permutations

An arrangement of objects or events in which the order is important is called a permutation.You can use a list to find the number of permutations of a group of objects.

Course 2

10-7 Permutations

A. In how many ways can you arrange the letters, A, B, and T ?

Additional Example 1A: Using a List to Find Permutations

Use a list to find the possible permutations.

There are 6 ways to order the letters.

A, B, T B, A, T T, A, B

A, T, B B, T, A T, B, A

Course 2

10-7 Permutations

B. If the order is chosen randomly, what is the probability that an arrangement will be a word?

Additional Example 1B: Using a List to Find Permutations

Of the 6 possible permutations, only 2 are words (B, A, T) and (T, A, B).

P(word) = number of word arrangementstotal number of arrangements

= 26

13

=

The probability that an arrangement is a word is 13 .

Course 2

10-7 Permutations

Try This: Example 1A

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A. In how many ways can you arrange the colors red, orange, blue?

Use a list to find the possible permutations.

There are 6 ways to order the colors.

red, orange, bluered, blue, orangeorange, red, blueorange, blue, redblue, orange, redblue, red, orange

Course 2

10-7 Permutations

Try This: Example 1B

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B. If the order is chosen randomly, what is the probability that the first letters of the color arrangements will be a word?

Of the 6 possible permutations, only three are words (orb), (bro), and (rob).

P(word) = number of word arrangementstotal number of arrangements

= 36

12

=

The probability that an arrangement is a word is 12 .

Course 2

10-7 Permutations

By making an organized list, you can findthe possible permutations as well as the number of permutations. You canuse the Fundamental Counting Principleto find only the number of permutations.

Course 2

10-7 Permutations

Mary, Rob, Carla, and Eli are lining up for lunch. In how many different ways can they line up? What is the probability that any one of the permutations will be chosen at random? Once you fill a position, you have one less choice for the next position.

Additional Example 2: Using the Fundamental Counting Principle to Find the Number of Permutations

There are 4 choices for the first position.

There are 3 remaining choices for the second position.

There are 2 remaining choices for the third position.There is one choice left for the fourth position.

4 · 3 · 2 · 1

There are 24 different ways the students can line up for lunch. The probability that any one of the permutations will be chosen is 1

24.

Multiply.= 24

Course 2

10-7 Permutations

Try This: Example 2

Find the number of ways you can rearrange the letters in the name Sam. What is the probability of any of the permutations being selected at random?

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There are 3 choices for the first position.There are 2 remaining choices for the second position.

There is one choice left for the third position.

3 · 2 · 1

There are 6 different ways the letters in the name Sam can be arranged. The probability that any one of the permutations will be chosen is . 1

6

.

Multiply.= 6

Course 2

10-7 Permutations

Multiplying 3 · 2 · 1 is called 3 factorialand is written as “3!” You can find the factorial of a whole number by multiplying all the whole numbers exceptzero that are less than or equal to the number.

3! = 3 · 2 · 1 = 6 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720

You can use factorials to find the number of permutations in a given situation.

Course 2

10-7 Permutations

In how many ways can Shellie line up 8 books on a shelf?

Additional Example 3: Using Factorials to Find the Number of Permutations

Number of permutations = 8!

= 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1

= 40,320

There are 40,320 different ways for Shellie to lineup 8 books on the shelf.

Course 2

10-7 Permutations

Try This: Example 3

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In how many ways can Sally line up 5 pictures on a desk?

Number of permutations = 5!

= 5 · 4 · 3 · 2 · 1

= 120

There are 120 different ways for Sally to lineup 5 pictures on a desk.

Course 2

10-7 Permutations

Lesson Quiz1. How many ways can Anna, Barbara, and Cara sit in a row?

2. If you choose one of the orders at random, what is the probability that Anna will be in the front?

Find the number of permutations of each situation.

3. How many different ways could 4 people enter a roller-coaster car?

4. How many different ways could 6 basketball players sit on the bench while waiting to be announced at the beginning of a game?

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Course 2

10-7 Permutations

Assignment

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