The task is to prove the following inequality:
$\begin{Bmatrix} mn\\ n \end{Bmatrix} \geqslant \frac{(mn)!}{(m!)^nn!}$ , where $m, n \in \mathbb{N_+}$
and to determine when the equality holds.
A simple substitution shows that the two sides are equal for $(m, n) = (1, 1) \wedge (m, n) = (1, n)$, but this observation doesn't bring me closer to the solution. The identities I know (which could help simplify any side of the inequality) don't seem of much use here (at least I don't see how to apply them), nor can I think of a good combinatorial approach.
Any idea how to tackle this problem?