Prove that this expression is always an integer

Question

Prove that $$\frac{(n^2)!}{(n!)^{n+1}}$$

is always an integer

From the counting problem permutation ways of $n$ times number from $1$ to $n$.

We can notice that $$\frac{(n^2)!}{(n!)^n}$$

is always an integer

Do they have a relationship?

Answer 1

Hint : Consider partitioning $n^2$ objects into $n$ blocks each of size $n$.