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Prove that (n !)^2<n^n(n !)<(2n)! for al...

Prove that `(n !)^2`<`n^n(n !)<(2n)!` for all positive integers n

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To prove that \( (n!)^2 < n^n (n!) < (2n)! \) for all positive integers \( n \), we will break the proof into two parts: first proving \( (n!)^2 < n^n (n!) \) and then proving \( n^n (n!) < (2n)! \). ### Step 1: Proving \( (n!)^2 < n^n (n!) \) 1. **Start with the definition of factorial**: By definition, \( n! = 1 \times 2 \times 3 \times \ldots \times n \). 2. **Consider the inequality**: ...
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