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) At least $3$ girls?

(ii) Exactly $3$ girls?

(iii) At most $3$ girls?

Answer:- (ii) Exactly $3$ girls?

This indicates to have to choose 3 girls from 4 also 4 boys from 9 to form a board of 7. This can be completed in

4C3​×9C4​

(ii) At least 3 girls

This indicates girls can either be 3 or 4

Consequently, first select 3 girls and 4 boys or 4 girls also 3 boys

Total ways = 4C3​×9C4​ + 4C4​×9C3​

(iii) At most 3 girls

This indicates we have to select 1 or 2 or 3 girls

Consequently, this can be completed in

4C0​×9C7​+4C1​×9C6​ + 4C2​×9C5​ + 4C3​×9C4​ ways.