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

(i) exactly 3 girls?

(ii) at least 3 girls?

(iii) at most 3 girls?

Solution:

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

(i) When the committee consists of exactly 3 girls:

Required number of ways $={ }^{4} C_{3} \times{ }^{9} C_{4}=\frac{4}{3} \times \frac{3}{2} \times \frac{2}{1} \times \frac{9}{4} \times \frac{8}{3} \times \frac{7}{2} \times \frac{6}{1}=504$

(ii) When the committee consists of at least 3 girls:

Required number of ways $={ }^{4} C_{3} \times{ }^{9} C_{4}+{ }^{4} C_{4} \times{ }^{9} C_{3}=504+84=588$

(iii) When the committee consists of at most 3 girls:

Required number of ways $={ }^{4} C_{0} \times{ }^{9} C_{7}+{ }^{4} C_{1} \times{ }^{9} C_{6}+{ }^{4} C_{2} \times{ }^{9} C_{5}+{ }^{4} C_{3} \times{ }^{9} C_{4}=36+336+756+504=1632$