Question

Using permutation or otherwise prove that $\frac{n^2!}{(n!{)}^n}$ is an integer, where n is a positive integer.

Answer 1

We have $n^2$ people, to be divided into $n$ teams, each of which has $n$ people.

Line up the $n^2$ people, put the first $n$ into a team, then the next $n$, and the next, and so on.

The no. of ways $n^2$ people can be divided into $n$ trams

$\frac{\left(n^2\right)!}{(n!{)}^n}$

No. of ways is always on integer.

Therefore $\frac{\left(n^2\right)!}{(n!{)}^n}$ is an integers