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Prove that ((n^(2))!)/((n!)^(n)) is a na...

Prove that `((n^(2))!)/((n!)^(n))` is a natural number for all n `in` N.

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To prove that \(\frac{(n^2)!}{(n!)^n}\) is a natural number for all \(n \in \mathbb{N}\), we will show that \((n^2)!\) is divisible by \((n!)^n\). ### Step-by-Step Solution: 1. **Understanding the Factorial**: We start with the expression \((n^2)!\). The factorial \(n^2\) is the product of all integers from 1 to \(n^2\): \[ (n^2)! = 1 \times 2 \times 3 \times \ldots \times (n^2) ...
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