Exercise 443Combinatorics › Counting principlesHigh schoolAdded 3 Nov 2026

Choosing a mixed committee

#combinations#committee#product rule

Statement

From $7$ men and $5$ women, how many committees of $4$ containing at least $2$ women can be formed?

Hint

Sum over the number of women ( $2$ , $3$ , $4$ ).

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Solution

Cases.
- $2$ women, $2$ men: $\binom{5}{2}\binom{7}{2}=10\cdot21=210$ .
- $3$ women, $1$ man: $\binom{5}{3}\binom{7}{1}=10\cdot7=70$ .
- $4$ women: $\binom{5}{4}=5$ .

Total. $210+70+5=285$ .

Conclusion. There are $285$ such committees.

Choosing a mixed committee

Statement

Solution

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