Question

Out of 6 ruling and 5 opposition party members, 4 are to be selected for a delegation. In how many ways can it be done so as to include at least one opposition member?

(a) 300

(b) 315

(c) 415

(d) 410

Answer 1

Correct option is (b) 315

4 members of the delegation can be selected in the following ways:

I. 1 opposition member and 3 Ruling party members, i.e., Number of ways of this selection = ⁵C₁ × ⁶C₃

II. 2 opposition members and 2 ruling party members, i.e., Number of ways of this selection = ⁵C₂ × ⁶C₂

III. 3 opposition members and 1 ruling party member, i.e., Number of ways of this selection = 5C3 × ⁶C₁

IV. 4 opposition members, i.e., Number of ways of this selection = ⁵C₄.

∴ Total number of ways for required selection = ⁵C₁ × ⁶C₃ + ⁵C₂ × ⁶C₂ + ⁵C₃ × ⁶C₁ + ⁵C₄

= \(5 \times \frac{6\times 5 \times 4}{3\times 2} + \frac{5\times 4}{2} \times \frac{6\times 5}2 + \frac {5\times 4}2 \times 6 + 5\)

= 100 + 150 + 60 + 5

= 315