Question

A committee of 5 persons is to be formed out of 6 gents and 4 ladies. In how many ways can this be done if at most two ladies are included?

(a) 168 

(b) 156 

(c) 186 

(d) 165

Answer 1

(c) 186

A committee of 5 persons, including at most two ladies can be constituted in the following ways : 

I. Selecting 5 gents only out of 6 gents. 

II. Selecting 4 gents out of 6 gents and 1 lady out of 4 ladies. 

III. Selecting 3 gents out of 6 gents and 2 ladies out of 4 ladies. 

∴ Total number of ways of forming the committee 

= ⁶C₅ + ⁶C₄ × ⁴C₁ + ⁶C₃ × ⁴C₂ 

= \(6+\frac{6\times5}{2}\times4+\frac{6\times5\times4}{3\times2}\times\frac{4\times3}{2}\) 

= 6 + 60 + 120 = 186.