Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 2 / a / Solution

Count pairs $(M,K)$ of subsets of an $n$-element candidate set with $K\subseteq M$, $|M|=m$, and $|K|=k$. Choosing the whole team $M$ first and then the special-role players $K$ gives $\left(\genfrac{}{}{0ex}{}{n}{m}\right)\left(\genfrac{}{}{0ex}{}{m}{k}\right)$. Alternatively choose $K$ first, then choose the other $m-k$ players from the $n-k$ candidates outside $K$, giving $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n-k}{m-k}\right)$. Each construction specifies the same pair uniquely. Thus double counting proves