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# 12.14: Combinations

Difficulty Level: Basic Created by: CK-12
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Jim and Maralie are in charge of tossing bracelets out into the audience for the school float during the town's Holiday Parade.  There are a total of four colors of bracelets:  orange, red, green, and blue.  They will reach into a basket and toss out three at a time.  Is this a combination or a permutation?

In this concept, you will learn the difference between combinations and permutations.

### Guidance

A combination is a collection of items in which order, or how the items are arranged, is not important. 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.

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.

Let's look at an example.

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?

First, write out a single order.

641, 224, 806

Next, rearrange the order. Did you change the outcome? If so, then order matters.

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

Order does NOT matter for this problem.

Here's another example.

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?

First, write out a single order.

red, blue

Next, 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.

### Guided Practice

Is the following 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?

First, write out a single order:

6, 14, 32

Next, rearrange the order:

14, 32, 6

Then, evaluate whether the order does matter given the situation.

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

### Examples

Write whether each example is a permutation or combination.

#### Example 1

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?

First, write out a single order:

Dog 1 - Leash 1

Dog 2 - Leash 2

Dog 3 - Leash 3

Next, rearrange the order:

Dog 1 - Leash 2

Dog 2 - Leash 1

Dog 3 - Leash 3

Then, evaluate whether the order does matter given the situation.

The answer is the order does not matter.  The dogs will get walked on leashes no matter the order.  This is a combination.

#### Example 2

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

First, write out a single order:

Horse 2, Horse 4, Horse 5

Next, rearrange the order:

Horse 4, Horse 2, Horse 5

Then, evaluate whether the order does matter given the situation.

The answer is the order does matter.  The order of the horses tells what place they come in for the race. This is a permutation.

#### Example 3

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

First, write out a single order:

Player 1, Player 3, Player 4, Player 5, Player 7

Next, rearrange the order:

Player 4, Player 3, Player 7, Player 1, Player 5

Then, evaluate whether the order does matter given the situation.

The answer is the order of choosing the teams does not matter.  This is a combination.

Do you remember Jim and Maralie's parade responsibility of tossing out bracelets from their school float during the Holiday Parade?  They will be tossing out three bracelets at a time taken from a basket with bracelets that are either orange red, green, or blue.  Is this an example of a permutation or a combination?

First, write out a single order:

red, green, blue

Next, rearrange the order:

green, red, blue

Then, evaluate whether the order does matter given the situation.

The answer is it does not make a difference in which color order the bracelets are tossed.  This is a combination.

### Explore More

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 A,B,C,D\begin{align*}A, B, C, D\end{align*}, and E\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?

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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