Question 183
Let $n \in \mathbb{N}$, $n \ge 2$, and three sets of real numbers $A, B, C$, pairwise disjoint, each of them having $n$ elements.
Let $a$ be the number of triples $(x, y, z) \in A \times B \times C$ for which $x \lt y \lt z$ and $b$ be the number of triples $(x, y, z) \in A \times B \times C$ for which $x \gt y \gt z$. Prove that $a - b$ is divisible by $n$.
Let $a$ be the number of triples $(x, y, z) \in A \times B \times C$ for which $x \lt y \lt z$ and $b$ be the number of triples $(x, y, z) \in A \times B \times C$ for which $x \gt y \gt z$. Prove that $a - b$ is divisible by $n$.