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Question

Difficulty: MediumCombinations

A school committee of 44 members is to be selected from 66 male teachers and 44 female teachers. If the committee must contain exactly 22 male teachers and 22 female teachers, how many different committees can be formed?

Answer: 90

Answer

The total number of different committees that can be formed is 9090.
Selecting 2 female teachers from 4 yields (42)=6\binom{4}{2} = 6 ways. Selecting 2 male teachers from 6 yields (62)=15\binom{6}{2} = 15 ways. By the multiplication principle, the total number of ways to form the committee is 6×15=906 \times 15 = 90.

Step-by-Step Solution

1
Find the number of ways to select 2 female teachers out of 4
\(\binom{4}{2} = 6\)
Selection order does not matter within the committee, so combination formula \(nCr\) applies.
2
Find the number of ways to select 2 male teachers out of 6
\(\binom{6}{2} = 15\)
Selection order does not matter within the committee, so combination formula \(nCr\) applies.
3
Multiply the possibilities for selecting male and female members
\(6 \times 15 = 90\)
According to the fundamental counting principle, independent group selections are multiplied.

Key Concept

Combinations with restricted subset selections (Product Rule of Counting)
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Topics

Combinations

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