Question 27
Let $p$ be a prime and $X$ be a finite set containing at least $p$ elements. A collection of mutually disjoint $p$-element subsets of $X$ is called a $p$-family. The empty collection itself is regarded as a $p$-family. Let $A$ (respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.