JEE Advance - Mathematics (2004 - No. 15)
Prove by permulation or otherwise $${{({n^2})!} \over {{{(n!)}^n}}}$$ is an integer $$(n \in {1^ + })$$.
This statement is false for all positive integers n.
The expression represents the number of ways to arrange n groups of n identical items, proving it's an integer.
The expression always results in a fraction that cannot be simplified to an integer.
This is a classic combinatorial problem involving arranging elements with repetitions, yielding an integer result.
The expression equals to 1 for all positive integers n.