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) atleast 3 girls ? 

(iii)atmost 3 girls

Answer 1

Number of girls = 4 

Number of boys = 9 

(i) Since the committee has to consists exactly 3 girls, the committee can consist of 3 girls and 4 boys. 

∴ 3 girls out of 4 can be selected in ⁴C₃ ways out of 4 boys out of 9 can be selected in ⁹C₄ ways

∴ Required number of ways = ⁴C₃ x ⁹C₄

= 4 x (9 x 8 x 7 x 6)/(4 x 3 x 2 x 1) = 504

(ii) Since the committee has to consist at least 3 girls, the committee can consists of 

(a) 3 girls and 4 boys 

(b) 4 girls and 3 boys 

(a) 3 girls and 4 boys can be selected in = ⁴C₃ x ⁹C₄ ways 

(b) 4 girls and 3 boys can be selected in ⁴C₄ x ⁹C₃ ways

∴ Required number of ways = ⁴C₃ x ⁹C₄ + ⁴C₄ x ⁹C₃ 

= 4 x 126 + 1 x 84 = 504 + 84 = 588 

(iii) In this case, commitee may consists of 3 girls, 2 girls ,1 girl or no girl.