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Combinations

Using lists and tree diagrams.

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Combinations

Let’s Think About It

License: CC BY-NC 3.0

As chairperson for the Performing Arts Center banquet, Tabitha has to help plan the meal. The caterer gave her a choice. She could choose a grilled chicken dinner, a roast beef dinner, or she could choose the buffet, which offered a selection of three entrees (grilled chicken, roast beef, and fish) and four side dishes. Tabitha thought the buffet was the best way to go, but before she agreed to it, she wanted to know how many different meal combinations they could make from the buffet if each meal consisted of one entrée and two sides.

In this concept, you will learn to find all possible combinations.

Guidance

Combinations are used for groupings in which order does not matter.

To figure out combinations, start by listing all of the possible options. Take a look at the following chart. You can use it to figure out all the possible 2-letter combinations you can make with the letters, A, B, C, and D.

A B C D
A
B
C
D

Next, go down the chart beginning with the A column and match A with each of the letters in the rows. For example, the first column would appear like this.

A B C D
A AA
B AB
C AC
D AD

Next, go down the chart beginning with the A column and match A with each of the letters in the rows. For example, the first column would appear like this.

You would do the same for each of the other columns. When you’re done, the chart would look like this.

A B C D
A AA BA CA DA
B AB BB CB DB
C AC BC CC DC
D AD BD CD DD

Then eliminate any duplicates. For example, AB and BA are duplicates because they are the same two letters but the order is reversed. So, BA can be eliminated.

A B C D
A AA BA CA DA
B AB BB CB DB
C AC BC CC DC
D AD BD CD DD

The highlighted terms are duplicates. The number of outcomes left is the combinations.

Here is a real-life example.

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, list all possible options beginning with Seth.

Seth can ride with Keith, Derek or Justin.

Then do the same with Keith and the others.

Keith can ride with Seth, Derek or Justin.

Derek can ride with Seth, Justin or Keith.

And Justin can ride with Seth, Derek or Keith.

Here are the possible combinations.

SK KS DS JS
SD KD DK JK
SJ KJ DJ JD

Next, cross out any duplicates. It does not matter if the letters are reversed. For example, because there is SD, you would cross out DS.

SK KS DS JS
SD KD DK JK
SJ KJ DJ JD

There are six different pair combinations left.

Guided Practice

How many combinations are possible in the following scenario?

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.

First, list all possible options beginning with red.

Red, Blue, Green Red, Blue, Yellow Red, Green, Yellow
Blue, Red, Green Blue, Red, Yellow Blue, Green, Yellow
Green, Red, Blue Green, Red, Yellow Green, Blue, Yellow
Yellow, Red, Blue Yellow, Red, Green Yellow, Blue, Green

Next, cross out any duplicates. Remember, order does not matter in combinations.

Red, Blue, Green Red, Blue, Yellow Red, Green, Yellow
Blue, Red, Green Blue, Red, Yellow Blue, Green, Yellow
Green, Red, Blue Green, Red, Yellow Green, Blue, Yellow
Yellow, Red, Blue Yellow, Red, Green Yellow, Blue, Green

The highlighted combinations are duplicates.

There are four possible combinations left.

The answer is four possible combinations.

Examples

In the following examples, find the possible combinations or answer questions about combinations.

Example 1

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

First, label each pair of sneakers.

A, B, C, D

Next, list all possible options beginning with sneakers A.

AB, AC, AD

Then, do the same for the others.

BA, BC, BD

CA, CB, CD

DA, DB, DC

Now, cross out the duplicates. Remember, in combinations, order does not matter.

AB, AC, AD

BA, BC, BD

CA, CB, CD

DA, DB, DC

There are 6 combinations left.

The answer is 6 combinations.

Example 2

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

First, label each of the 5 colors or use the name of the color. For simplicity, let’s use letters to denote colors.

A, B, C, D, E

Next, list all possible 3-colors options beginning with A.

ABC, ABD, ABE, ACB, ACD, ACE, ADB, ADC, ADE, AEB, AEC, AED

Then, do the same for the others.

BAC, BAD, BAE, BCA, BCD, BCE, BDA, BDC, BDE, BEA, BEC, BED

CAB, CAD, CAE, CBA, CBD, CBE, CDA, CDB, CDE, CEA, CEB, CED

DAB, DAC, DAE, DBA, DBC, DBE, DCA, DCB, DCE, DEA, DEB, DEC

EAB, EAC, EAD, EBA, EBC, EBD, ECA, ECB, ECD, EDA, EDB, EDC

Now, cross out the duplicates. Remember, in combinations, order does not matter.

ABC, ABD, ABE, ACB, ACD, ACE, ADB, ADC, ADE, AEB, AEC, AED

BAC, BAD, BAE, BCA, BCD, BCE, BDA, BDC, BDE, BEA, BEC, BED

CAB, CAD, CAE, CBA, CBD, CBE, CDA, CDB, CDE, CEA, CEB, CED

DAB, DAC, DAE, DBA, DBC, DBE, DCA, DCB, DCE, DEA, DEB, DEC

EAB, EAC, EAD, EBA, EBC, EBD, ECA, ECB, ECD, EDA, EDB, EDC

There are 10 combinations left.

The answer is 10 combinations.

Example 3

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

The answer is false. Order does not make a difference in a combination.

Follow Up

License: CC BY-NC 3.0

Remember Tabitha and her meal decision for the Honor Society’s banquet?

Tabitha expected about 125 people for the Performing Arts Center banquet, and she decided a buffet-style dinner as the best way to go. The caterer said there would be three entrées and four sides on the buffet. A dinner consisted of one entrée and two sides. Tabitha sat down and began figuring out how many possible dinner combinations could be made from the buffet.

How can you help Tabitha find all the possible combinations?

First, list all possible options for each entree.

Entrée 1 Entrée 2 Entrée 3

Next, list the side dish combinations for each entrée. Sides are labeled A, B, C, D.

Entrée 1 Entrée 2 Entrée 3
AB AB AB
AC AC AC
AD AD AD
BA BA BA
BC BC BC
BD BD BD
CA CA CA
CB CB CB
CD CD CD
DA DA DA
DB DB DB
DC DC DC

Then, cross out any duplicates. Remember, order does not matter in combinations.

Entrée 1 Entrée 2 Entrée 3
AB AB AB
AC AC AC
AD AD AD
BA BA BA
BC BC BC
BD BD BD
CA CA CA
CB CB CB
CD CD CD
DA DA DA
DB DB DB
DC DC DC

The highlighted combinations are duplicates.

There are 18 possible combinations left.

The answer is the buffet offers 18 possible meal combinations.

Video Review

https://www.youtube.com/watch?v=bCxMhncR7PU

https://www.youtube.com/watch?v=SGn1913lOYM

Explore More

Determine all possible combinations to the scenarios below and answer any questions asked about combinations or situations.

  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. 13 ice cream flavors taken two at a time.
  11. 13 ice cream flavors taken three at a time.
  12. 13 ice cream flavors taken four at a time.
  13. 10 children arranged in groups of five
  14. 10 children arranged in groups of four
  15. 10 children arranged in groups of three

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.18. 

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Vocabulary

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.

Outcome

An outcome of a probability experiment is one possible end result.

Permutation

A permutation is an arrangement of objects where order is important.

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.

Tree Diagram

A tree diagram is a visual way of showing options and variables. The lines of a tree diagram look like branches on a tree.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

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