Question

A committee of 4 is to be selected from amongst 5 boys and 6 girls. In how many ways can this be done so as to include (i) exactly one girl, (ii) at least one girl?

Answer 1

(i) In this case we have to select one girl out of 6 and 3 boys out of 5. 

The number of ways of selecting 3 boys = ⁵C₃ = ⁵C₂ = 10. 

The number of ways of selecting one girl = ⁶C₁ = 6. 

∴ The required committee can be formed in 6 × 10 = 60 ways. 

(ii) The committee can be formed with (a) one boy and three girls, or (b) 2 boys and 2 girls, 

or (c) 3 boys and one girl, or (d) 4 girls alone. 

The committee can be formed in (a) ⁵C₁ × ⁶C₃ ways ; 

The committee can be formed in (b) ⁵C₂ × ⁶C₂ ways ; 

The committee can be formed in (c) ⁵C₃ × ⁶C₁ ways ; 

The committee can be formed in (d) ⁶C₄ ways. 

Hence the required number of ways of forming the committee 

= ⁵C₁ × ⁶C₃ + ⁵C₂ × ⁶C₂ + ⁵C₃ × ⁶C₁ + ⁶C₄ = 100 + 150 + 60 + 15 = 325 ways. 

(ii) Method II. Required ways = (Committees of 4 out of 11 without any restriction) – (Committees in which no girl is included) = ¹¹C₄ – ⁵C₄ = 325.