proving that $\frac{(n^2)!}{(n!)^n}$ is an integer How to prove that $$\frac{(n^2)!}{(n!)^n}$$ is always a positive integer when n is also a positive integer. NOTE i want to prove it without induction. I just cancelled $n!$ and split term which are $n^2-(a^2)=(n-a)(n+a)$ where a is a perfect square. nothing more i could do. 

$S_n \times S_n \times \cdots \times S_n$ is a subgroup of $S_{n^2}$.
The number in question is the index of that subgroup and thus an integer by Lagrange's theorem.
