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Combinations

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Combinations

Have you ever tried to organize a wardrobe or your closet? Look at this dilemma.

James needs to choose a 2-color combination for his intramural team t-shirts. How many different 2-color combinations can James make out of red, blue, and yellow?

To figure this out, you will need to know about combinations. Pay attention to this Concept and you will be able to solve this problem by the end of the Concept.

Guidance

When you have a combination, order does not matter. The ice cream cones were a good example. It did not matter what the order was of the flavors or the cones. We just wanted to know how many different possible cones could be created.

We can find all of the possible combinations when working with examples.

How do we do that?

We work on figuring out combinations by listing out all of the possible options. Then we eliminate any duplicates and the number of outcomes left is our answer.

Seth, Keith, Derek and Justin want to go on the bumper cars. They can only ride in pairs. How many different paired combinations are possible given these parameters?

To start, we list out all possible options beginning with Seth. Seth can ride with Keith Derek or Justin. Keith can ride with Seth, Derek or Justin. Derek can ride with Seth, Justin or Keith. Justin can ride with Seth, Derek or Keith.

Here are the possible combinations.

&\text{SK} && \text{KS} && \text{DS} && \text{JS}\\&\text{SD} && \text{KD} && \text{DK} && \text{JK}\\&\text{SJ} && \text{KJ} && \text{DJ} && \text{JD}

Next we cross out any duplicates.

&\text{SK} && \text{\bcancel{KS}} && \text{\bcancel{DS}} && \text{\cancel{JS}}\\&\text{SD} && \text{KD} && \text{\cancel{DK}} && \text{\cancel{JK}}\\	&\text{SJ} && \text{KJ} && \text{DJ} && \text{\cancel{JD}}

There are six different pair combinations.

What if order had made a difference? What if we had wanted to count each person if they sat in a different seat? What would have happened then?

That is where our next way of figuring outcomes comes in. It is called permutations.

Find combinations or answer questions about combinations.

Example A

Kyle has four different pairs of sneakers. He can only bring two pairs to camp. How many different combinations can he make?

Solution: 6 combinations

Example B

How many different combinations can you choose from five colors taken three at a time?

Solution: 10 combinations

Example C

True or false. In a combination, order makes a difference.

Solution: False. Order does not make a difference in a combination.

Here is the original problem once again.

James needs to choose a 2-color combination for his intramural team t-shirts. How many different 2-color combinations can James make out of red, blue, and yellow?

One way to find the number of combinations is to make a tree diagram. Here, if red is chosen as one color, that leaves only blue and yellow for the second color.

The diagram shows all 6 permutations of the 3 colors. But wait–since we are counting COMBINATIONS here order doesn’t matter.

So in this tree diagram we will cross out all outcomes that are repeats. For instance, the first red-blue is no different from blue-red, so we’ll cross out blue-red.

There are 3 combinations that are not repeats.

Vocabulary

Here are the vocabulary words in this Concept.

Probability
the chances or likelihood that an event will happen.
Outcome
the end result
Tree Diagram
a visual way of showing options and variables in an organized way. The lines of a tree diagram look like branches on a tree.
Combination
an arrangement of options where order does not make a difference.

Guided Practice

Here is one for you to try on your own.

The sixth grade class voted on colors for the school flag. The top choices were red, blue, green and yellow. The students can only choose three colors. How many combinations are possible?

Answer

You can list out all of the options to figure this out.

red, blue, green

red, blue, yellow

red, yellow, green

blue, green, yellow

There are four possible combinations.

Video Review

Here are videos for review.

Khan Academy, Combinations

James Sousa, Combinations

Practice

Directions:Figure out the possible combinations for each situation or answer questions about combinations.

1. 5 colors taken four at a time

2. 6 colors taken two at a time

3. 6 colors taken four at a time

4. 6 colors taken three at a time

5. 7 dogs walked two at a time

6. 7 dogs walked three at a time

7. 7 dogs walked four at a time

8. 7 dogs walked five at a time

9. Look back at the dog problems. Do you see a pattern?

10. 33 ice cream flavors taken two at a time.

11. 33 ice cream flavors taken three at a time.

12. 33 ice cream flavors taken four at a time.

13. 16 children arranged in groups of five

14. 16 children arranged in groups of four

15. 16 children arranged in groups of three

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