IMC 2021 · Problem 2

Let $n$ and $k$ be fixed positive integers, and let $a$ be an arbitrary non-negative integer. Choose a random $k$-element subset $X$ of $\{1, 2, \ldots, k + a\}$ uniformly (i.e., all $k$-element subsets are chosen with the same probability) and, independently of $X$, choose a random $n$-element subset $Y$ of $\{1, \ldots, k + n + a\}$ uniformly.

Prove that the probability

$$\mathsf{P}\Big(\min(Y) > \max(X)\Big)$$

does not depend on $a$.