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Question
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: atmost 3 girls?
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Solution
If maximum 3 girls are to be included in the committee, then the committees will be formed as follows:
- No girls and 7 boys
- 1 girl and 6 boys
- 2 girls and 5 boys
- 3 girls and 4 boys
Hence, the total committees formed = 4C0 x 9C7 + 4C1 x 9C6 + 4C2 x 9C5 + 4C3 x 9C4
= 1 x 9C2 + 4C1 x 9C3 + 4C2 x 9C4 + 4C1 x 9C4
= 1 x `(9 xx 8)/(1 xx 2) + 4/1 xx (9 xx 8 xx 7)/(1 xx 2 xx 3) + (4 xx 3)/(1 xx 7) xx (9 xx 8 xx 7 xx 6)/(1 xx 2 xx 3 xx 4) + 4/1 xx (9 xx 8 xx 7 xx 6)/(1 xx 2 xx 3 xx 4)`
= 1 x 36 + 4 x 84 + 6 x 126 + 4 x 126
= 36 + 336 + 126 x (6+ 4)
= 372 + 1260
= 1632
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