IMC 2021 Problem 2

Let $n$ and $k$ be fixed positive integers , and $a$ be arbitrary nonnegative integer . Choose a random $k$ -element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of $X$ , choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly. Prove that the probability $P\left( \text{min}(Y)>\text{max}(X)\right)$ does not depend on $a$ .