12.1.1. Fundamental Counting Principle, Permutations, and Combinations

A teacher is supposed to form a 6-student committee that consists of 4 boys and 2 girls. How many ways can the committee be formed from a group of 8 boys and 7 girls?

• A
1 470
• B70
• C21
• D91

Example

A teacher is supposed to form a 6-student committee that consists of 4 boys and 2 girls. How many ways can the committee be formed from a group of 8 boys and 7 girls?

Solution

This problem needs to be solved in two parts. First, the number of ways 4 boys can be selected from 8 needs to be determined. A combination is used because the order in which the boys are chosen does not matter.

The formula for combination is . For the boys, and . Substituting these values into the formula, we have

There are 70 ways 4 boys can be chosen from 8.

Next, the number of ways 2 girls can be selected from 7 girls needs to be determined. Again, a combination is used with and . Substituting these values into the formula, we have

There are 21 ways 2 girls can be chosen from 7.

To determine the number of ways the committee can be formed, multiply the number of ways the boys can be chosen by the number of ways the girls can be chosen.

There are ways the committee can be formed.

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