US-TST 2024 Problem 3

Let $n > k\geq 1$ be integers and let $p$ be a prime dividing ${\binom{n}{k}}$ . Prove that the $k$ -element subsets of $\{1,\ldots ,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.