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I'm trying to make sense of a proof and there is only one step I don't understand: the first one

$\prod_{k=0}^n \binom{n}{k} = \frac{n!^{n+1}}{\prod_{k=0}^n (k!)^2}$

Any help would be much appreciated

Here's the whole proof: https://www.cut-the-knot.org/arithmetic/algebra/HarlanBrothers.shtml

This is all for a presentation on pascal's triangles and I'm providing proofs for various interesting properties.

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$$\prod_{k=0}^n \binom{n}{k} = \prod_{k=0}^n \frac{n!}{k! (n-k)!}$$

Now you can see that $\prod_{k=0}^n(n-k)!=\prod_{k=0}^nk!$ as for $0\leq k \leq n$ $n-k$ spans $0..n$.

Substitute in the original equation and you have your result.

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