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) exactly 3 girls?

(ii) at least 3 girls?

(iii) at most 3 girls?

A committee of 7 has to be formed from 9 boys and 4 girls.

I. Exactly 3 girls: If there are exactly 3 girls in the committee, then there must be 4 boys, and the ways in which they can be chosen is

= ⁴C₃ ⁹C₄

= 504 ways

II. At least 3 girls: Here the possibilities are

(i) 3 girls and 4 boys and

(ii) 4 girls and 3 boys.

the number of ways they can be selected

= ⁴C₃ ⁹C₄ + ⁴C₄ ⁹C₃

₌ 588

III. At most 3 girls:

(i) 7 boys but no girls

(ii) 6 boys and 1 girl

(iii) 5 boys and 2 girls &

(iv) 4 boys and 3 girls.

And the number of their selection is

= ⁴C₃ ⁹C₄ + ⁴C₂ ⁹C₅ + ⁴C₁ ⁹C₆ + ⁴C₀ ⁹C₇

= 1584 ways.