# 12.14: Combinations

Difficulty Level: Basic Created by: CK-12
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Practice Combinations

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Have you ever been on a decorating committee?

The decorating committee is getting the stage ready for the Talent Show. There was a bunch of different decorating supplies ordered, and the students on the committee are working on figuring out the best way to decorate the stage.

They have four different colors of streamers to use to decorate.

Red

Blue

Green

Yellow

“I think four is too many colors. How about if we choose three of the four colors to decorate with?” Keith asks the group.

“I like that idea,” Sara chimes in. “How many ways can we decorate the stage if we do that?”

The group begins to figure this out on a piece of paper.

Is this a combination or a permutation?

Use what you will learn in this Concept to answer this question.

### Guidance

In the last Concept, you saw that order is important for some groups of items but not important for others. Consider a list of three words: HOPS, SHOP, and POSH.

• For the spelling of each individual word, order is important. The words HOPS, SHOP, and POSH all use the same letters, but spell out very different words.
• For the list itself, order is not important. Whether the words are presented in one order–such as HOPS, SHOP, POSH, or another order, such as SHOP, POSH, HOPS, or a third order, such as POSH, HOPS, SHOP–makes no difference. As long as the list includes all 3 words, the order of the 3 words doesn’t matter.

A combination is a collection of items in which order, or how the items are arranged, is not important. The collection of one order of the items is not functionally different than any other order.

Combinations and permutations are related. To solve problems in which order matters, you use permutations. To solve problems in which order does NOT matter, use combinations.

Take a look at this situation.

The winning 3-digit lottery numbers are drawn from a drum as 641, 224, and 806. Does order matter in the way the three winning numbers are drawn?

Step 1: Write out a single order.

641, 224, 806

Step 2: Now rearrange the order. Did you change the outcome? If so, then order matters.

\begin{align*}224, 806, 641\Longleftarrow \end{align*} different order, same 3 winning numbers

Order does NOT matter for this problem. Use combinations.

Write the difference between combinations and permutations down in your notebook.

Here is another one.

A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 1 marble, then return the marble to the bag and draw out a second marble?

Step 1: Write out a single order.

red, blue

Step 2: Now rearrange the order. Did you change the outcome? If so, then order matters.

blue, red \begin{align*}\Longleftarrow \end{align*} different order, meaning is DIFFERENT

Order DOES matter for this problem. Use permutations.

Write whether you would use combinations or permutations for each example.

#### Example A

Cesar the dog-walker has 5 dogs but only 3 leashes. How many different ways can Cesar take a walk with all 3 dogs at once?

Solution: Combination

#### Example B

Five different horses entered the Kentucky Derby. How many different ways can the horses finish the race?

Solution: Combination

#### Example C

How many different 5-player teams can you choose from a total of 8 basketball players?

Solution: Combination

Here is the original problem once again.

The decorating committee is getting the stage ready for the Talent Show. There was a bunch of different decorating supplies ordered, and the students on the committee are working on figuring out the best way to decorate the stage.

They have four different colors of streamers to use to decorate.

Red

Blue

Green

Yellow

“I think four is too many colors. How about if we choose three of the four colors to decorate with?” Keith asks the group.

“I like that idea,” Sara chimes in. “How many ways can we decorate the stage if we do that?”

The group begins to figure this out on a piece of paper.

Combinations are arrangements where order does not make a difference. The decorating committee is selected three colors from the possible four options. Therefore, the order of the colors doesn’t matter. This is a combination.

### Vocabulary

Here are the vocabulary words in this Concept.

Combination
an arrangement of objects or events where order does not matter.
Permutations
an arrangement of objects or events where the order does matter.

### Guided Practice

Here is one for you to try on your own.

Is this a permutation or a combination?

Mario’s gym locker uses the numbers 14, 6, and 32. How many different arrangements of the three numbers must Mario try to be sure he opens his locker?

There is only one way that the numbers can be arranged to open Mario's locker. This is a permutation because order does matter.

### Video Review

Here is a video for review.

### Practice

Directions: Write whether you are more likely to use permutations or combinations for each of the following examples.

1. A bag has 4 marbles: red, blue, yellow, and green. In how many different ways can you reach into the bag and draw out 2 marbles at once and drop them in a cup?

2. A bag contains 5 slips of paper with letters \begin{align*}A, B, C, D\end{align*}, and \begin{align*}E\end{align*} written on them. Pull out one slip, mark down the letter and replace it in the bag. Do this 3 times so you have written 3 letters. How many different ways can you write the 3 letters?

3. Eight candidates are running for the 4-person Student Council. How many different Student Councils are possible?

4. Mark's gym locker uses the numbers 24, 36, and 2. How many different arrangements of the three numbers must Mark try to be sure he opens his locker?

5. Five horn players are running for 2 seats in a jazz band. How many different ways can the two horn players be chosen?

Directions: Identify each situation as a permutation or combination.

6. The medals awarded for a swimming meet.

7. The order that you put toppings on a pizza.

8. The numbers that open a combination lock.

9. The way that you organize your clothing in a drawer.

10. A baseball line up according to ability.

11. A group of people organized randomly.

12. Walking five dogs three at a time.

13. Walking five dogs with a specific order.

14. Nine candidates are running for the 5-person Student Council. How many different Student Councils are possible?

15. Twelve candidates are running for the 6-person Student Council. How many different Student Councils are possible?

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