# 12.12: Permutations

**At Grade**Created by: CK-12

**Practice**Permutations

Naima and Michael are in charge of staffing the middle school Sock Hop. They begin with the five 6th-graders who have volunteered to work: Cindy, Justin, Willy, Donna, and Angelle. They are trying to figure out the order of assignment. Does order matter with regard to this situation?

In this concept, you will learn about permutations.

### Permutations

You can make all kinds of combinations. Let’s say that you are making a pizza with pepperoni, mushrooms and peppers. It doesn’t matter which order you put the toppings on the pizza. You will still have the same pizza.

Sometimes, order does make a difference. When you arrange different objects or events and order is important, this is called a **permutation**.

Consider the word CAT. Clearly, order is important when you spell a word. You can write all of the correct letters, but if you don’t put them in the correct order, you don’t spell CAT. For example, here are some orders of C, A, and T that *don’t* spell CAT;

\begin{align*}& \text{Incorrect orders:} && ACT, ATC, CTA, TAC, TCA\\ & \text{Correct orders:} && CAT\end{align*}

This is a situation where order makes a difference.

Let's look at an example.

Tomás wants to know how many 3-digit numbers he can write using the digits 7, 8, and 9 without repeating any of the digits. Does order matter for this problem?

First, write out a single order: 789

Next, rearrange the order: 798

Then decide if order matters: This is a different number from the original number, so order does matter. Each arrangement of digits is a different permutation.

### Examples

#### Example 1

Earlier, you were given a problem about Naima and Michael, who are assigning job duties for the middle school dance.

They are assigning entrance door duty, beginning with the 6th-graders: Cindy, Justin, Willy, Donna, and Angelle. Does the order in which they are assigned to perform the duty matter?

First, write out a single order:

Cindy, Justin, Willy, Donna, Angelle

Next, rearrange the order:

Willy, Donna, Cindy, Angelle, Justin

Then, decide if the order of the assignment matters:

The answer is the order does not matter; the duties will be covered by a 6th-grader no matter what the order. This is not a permutation.

#### Example 2

Henry has a combination lock. He can program the combination of his lock using the numbers 1, 5 and 18. Henry can only choose one arrangement of the numbers. Is this a permutation? Why?

First, write out a single order: 1, 5, 18

Next, rearrange the order: 5, 18, 1

Then, decide if order matters in this situation.

The answer is order does matter in this situation since there is only one way to order the numbers that will operate the lock. This is a permutation.

**Write whether or not order is important for each scenario and why.**

#### Example 3

The number of 3-letter words Brenda can write using the letters \begin{align*}E, T, A\end{align*} without repeating letters.

First, write out a 3-letter word.

T-E-A

Next, rearrange the lineup:

A-T-E

Then, decide if you change the outcome by rearranging the order. If so, then order matters.

The answer is T-E-A is a different word from A-T-E, so order does matter. This is an example of a permutation.

#### Example 4

Five different cars entered the race painted the following colors: red, orange, blue, white, purple? In how many different ways can the cars finish the race?

First, write out a single lineup:

purple, white, red, orange, blue

Next, rearrange the lineup:

red, blue, orange, purple, white

Then, decide if you change the outcome? If so, then order matters.

The answer is the change in order results in a different winner, so order does matter. This is a permutation.

#### Example 5

At the breakfast buffet you can take any three of the following: eggs, pancakes, potatoes, cereal, waffles. How many different 3-item breakfasts can you get?

First, write out a single order:

eggs-pancakes-potatoes

Next, rearrange the order:

potatoes-eggs-pancakes

Then decide if order matters:

The answer is in either order, you will get the same food. So order does not matter. This is not a permutation.

### Review

Decide whether or not order matters for each of the following scenarios. Briefly explain your reasoning.

- Doug is going to use the following 5 letters to create his new 3-letter computer password: B, F, G, L, and T. How many different passwords can he create if he doesn’t repeat any letters?
- Violin players in the orchestra include Jerry, Kerry, Barry, Mary, Sherry, Harry, Terri, and Perry. How many different 3-person trios can you make?
- The 3 different numbers for Arun’s lock are 14, 35, and 20. How many different combinations must Arun try before he’ll be sure he can open his lock?
- Mr. Chen has decided that he’s going to give Nikki, Mickey, and Hickey awards for the essay contest. What he doesn’t know is who will get 1st prize, 2nd prize, and 3rd prize. How many different ways can Mr. Chen give out the prizes?
- Five candidates are running for 2 student senate seats: Bo, Jo, Mo, Zo, and Ro. How many different pairs of senators can there be?
- Five skaters are competing in the County Championship Finals: Miller, Diller, Hiller, Giller, and Stiller? How many different ways can they finish first, second, third, fourth, and fifth?

Find the number of permutations for each problem.

- How many 3-letter words can Brenda write using the letters A, B, and C without repeating any of the letters?
- How many 4-digit numbers can Brenda write using the digits 1, 3, 5, 7 without repeating any of the digits?
- Doug, Eileen, Francesca, and Garth all entered the swimming race. In how many different orders can the four racers finish?
- Miguel is serving soup, salad, pasta, and fish for dinner. In how many different orders can he serve the four dishes?
- Mike has 4 different playing cards: Ace, King, Jack, and Ten. How many different 4-card arrangements can he make?
- Marlena strung 5 charms on a bracelet–a star, a fish, a diamond, a moon, and a baby shoe. Into how many different orders can she arrange the 5 charms?
- Michelle forgot her 6-letter computer password. She knows she used the letters H, I, J, K, L, and M in the password and that she didn’t repeat any of the letters. How many different passwords must she try before she is sure to hit the correct one?
- Seven skaters are competing in the County Championship Finals. In how many different orders can they finish first, second, third, fourth, fifth, sixth, and seventh?
- If there are eight skaters competing, how does this change the number of permutations?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.12.

### Resources

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In this concept, you will learn about permutations.

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