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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 at least 3 women?
Solution
Number of men = 8
Number of women = 4
Number of peoples in the committee = 7
At least 3 women
The 7 members committee must contain at least 3 women
∴ We have the following possibilities
(i) 4 women + 3 men
(ii) 3 women + 4 men
Case (i): 4 women + 3 men
The number ways of selecting 4 women .from
4 women is = 4C4 = 1 way
The number of ways of selecting 3 men from 8 men = 8C3
= `(8!)/(3! xx (8 - 3)!)`
= `(8!)/(3! xx 5!)`
= `(8 xx 7 xx 6 xx 5!)/(3!xx 5!)`
= `(8 xx 7 xx 6)/(3 xx 2 xx 1)`
= 8 × 7
= 56
∴ The total number of ways = 1 × 56 = 56
Case (ii): 3 women + 4 men
The number of ways of selecting 3 women from 4 women is = 4C3
The number of ways of selecting 4 men from 8 men is = 8C4
∴ The total number of ways = 4C3 × 8C4
= `(4!)/(3! xx (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
∴ The required number of ways of forming the committee = 56 + 280 = 336
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