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:
(i) exactrly 3 girls?
(ii) at least 3 girls
(ii) almost 3 girls?

Explanation

(i) If there are 3 girls in the committee then there will be 4 boys. So, no. of committee from by selecting 4 boys from 9 boys and 3 girls from 4 girls
`= .^(9)C_(4) xx .^(4)C_(3)`
`=(9 xx 8 xx 7 xx 6)/(4 xx 3xx 2xx1) xx 4`
`= 504`
(ii) If there are at leasty 3 girls in the committee then the committee will be selected in the following forms
(a) 3 girls, 4 boys
(b) 4 girls, 3 boys.
Therefore, required number of committee
`= .^(4)C_(3) xx .^(9)C_(4) + .^(4)C_(4) xx .^(9)C_(3)`
`= 4 xx (9 xx 8 xx 7 xx 6)/(4 xx 3 xx 2 xx 1) +1 xx (9 xx 8 xx 7)/(3 xx 2 xx 1)`
`= 504 +84 = 588`
(iii) If there are almosy 3 girls in the committee then the committee will be selected in the following forms:
(a) 0 girls, 7 boys (b) 1 girl, 6 boys
(c) 2 girls, 5 boys (d) 3 girls, 4 boys
The number of committee
`= .^(4)C_(0) xx .^(9)C_(7) +.^(4)C_(1) xx .^(9)C_(6) +.^(4)C_(2) xx .^(9)C_(5) +.^(4)C_(3) xx .^(9)C_(4)`
`=1 xx (9 xx 8)/(2 xx 1) + 4 xx (9 xx 8 xx 7)/(3 xx 2 xx 1) +(4 xx 3)/(2 xx 1) xx (9 xx 8 xx 7 xx 6)/(4 xx 3xx 2xx 1)+4 xx (9 xx 8 xx 7 xx 6)/(4 xx 3 xx 2xx 1)`
`= 36 +336 +756 +504 = 1632`