Q:

Help on this step: $\prod_{k=0}^n \binom{n}{k} = \frac{n!^{n+1}}{\prod_{k=0}^n (k!)^2}$

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. 

A:

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