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# Averages, Probability, and Combinations for CBEST Exam Study Guide (page 5)

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Updated on Mar 29, 2011

### Combinations

Combination problems require you to make as many groups as possible given certain criteria. There are many different types of combination problems, so these questions need to be read carefully before attempting to solve them. One of the easiest ways to make combination problems into CBEST points is to make a chart called a tree diagram and list all the possibilities in a pattern. The following sample question is a typical CBEST combination problem.

#### Sample Combination Question

1. Shirley had three pairs of slacks and four blouses. How many different combinations of one pair of slacks and one blouse could she make?
1. 3
2. 4
3. 7
4. 12
5. 15

To see this problem more clearly, you may want to make a tree diagram:

Each pair of slacks can be matched to four different blouses, making four different outfits for each pair of the three pairs of slacks, 3 × 4, for a total of 12 possible combinations.

#### Sample Combination Question

1. Five tennis players each played each other once. How many games were played?
1. 25
2. 20
3. 15
4. 10
5. 5

This combination problem is a little trickier in that there are not separate groups of items as there were for the slacks and blouses. This question involves the same players playing each other. But solving it is not difficult. First, take the total number of players and subtract one: 5 – 1 = 4. Add the numbers from 4 down: 1 + 2 + 3 + 4 = 10. To learn how this works, take a look at the following:

Letter the five players from A to E:

• A plays B, C, D, and E (4 games)
• B has already played A, so needs to play C, D, E (3 games)
• C has already played A and B, so needs to play D, E (2 games)
• D has already played A, B, and C, so needs to play E (1 game)
• E has played everyone

Adding up the number of games played (1 + 2 + 3 + 4) gives a total of 10, choice d.

This same question might be asked on the CBEST using the number of games five chess players played or the number of handshakes that occur when five people shake hands with each other once.

### Other Combination Problems

Although the previous combination problems are the most common, other kinds of problems are possible. The best way to solve other combination problems is to make a list. When you notice a pattern, stop and multiply. For example, if you're asked to make all the possible combinations of three letters using the letters A through D, start with A:

There seem to be 16 possibilities that begin with A, so there are probably 16 that begin with B and 16 each that begin with C and D, so multiplying 16 × 4 will give you the total possible combinations: 64.