Question

A consulting company needs five consultants to work on a project. There are 15 consultants available, of which two are married to each other. The company’s policy is that the married couple should not work together in the project. What is the probability of choosing a team of 5 consultants such that the company’s policy is satisfied?

Answer 1

Here there are 15 consultants available and 5 have to be selected.

Total number of ways

$C_5^{15} = \frac{15!}{5!(15-5)!}\\ = \frac{11 \times 12 \times 13 \times 14 \times 15}{2 \times 3 \times 4 \times 5} \\ = 3003$

Here if we don't follow company policy we will employ both of the married couple

Total number of ways then $= 13C^{13}_3 \times 1$

$= \frac{13!}{3!(13-3)!} \\ = \frac{11 \times 12 \times 13}{2 \times 3} \\ = 286$

So, number of ways when we follow company policy $= 3003 - 286 = 2717$

P(We will satisfy company policy) $= \frac{2717}{3003} = 0.9048$