A committee of 5 persons is to be formed out of 6 gents and 4 ladies.

1.	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» (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.

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»

(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.