Let $m\ge1$, $n\ge1$, and $k\ge1$ be integers with $k\le m + n$. Consider a set $P$ consisting of $m$ men and $n$ women. We choose a uniformly random $k$-element subset $Q$ of $P$. Consider the random variables
- $X$ = the number of men in the chosen subset $Q$,
- $Y$ = the number of women in the chosen subset $Q$,
- $Z = X − Y$.
Determine the expected value ${\rm E}(X)$.
Prove that ${\rm E}(Z) = k\dfrac{m - n}{m + n}$.