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Question
A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of exactly 3 women?
Solution
Number of men = 8
Number of women = 4
Number of peoples in the committee = 7
Exactly 3 women
In a 7 member committee, women must be 3. Therefore, the remaining 4 must be men.
The number of ways of selecting 3 women from 4 women = 4C3
The number of ways of selecting 4 men from 8 men = 8C4
∴ The total number of ways of selection is = 4C3 × 8C4
= `(4!)/(3!(4 - 3)!) xx (8!)/(4!(8 - 4)!)`
= `(4!)/(3! xx 1!)xx (8)/(4! xx 4!)`
= `(4 xx 3!)/(3!) xx (8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 4!)`
= `4 xx (8 xx 7 xx 6 xx 5)/(4!)`
= `(4 xx 8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1)`
= 8 × 7 × 5
= 280
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