Problem - 4363
Show that for any positive integer $n$, the value of $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer.
Let's consider the following counting problem: how many different ways to distribute $n^2$ distinguishable balls equally among $n$ indistinguishable boxes. The answer is $$\frac{1}{n!}\cdot\frac{(n^2)!}{n!\cdot n!\cdots n!}=\frac{(n^2)!}{(n!)^{n+1}}$$
Therefore, this value must be an integer.