Question

A committee of 4 men and 6 women is to be selected from 7 men and 8 women. If there is a married couple among the 15 people, in how many ways can the committee be selected so that the couple are automatically in the committee?

Answer 1

As the couple are automatically in the committee, we need to count the number of the ways to be selected 3 men from 6 and 5 women from 7. The order does not matter, so we count the number of combinations without repetition:

$\begin{pmatrix} 6 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 7 \\ 5 \end{pmatrix}=\cfrac{6!}{3!\cdot(6-3)!}\cdot\cfrac{7!}{5!\cdot(7-5)!}=\\ =\cfrac{4\cdot5\cdot6}{2\cdot3}\cdot\cfrac{6\cdot7}{2}=420.$