ILL 1974 Problem 7

Let $p$ be a prime number and $n$ a positive integer. Prove that the product $${N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]$$ Is a positive integer that is not divisible by $p.$