Factorial$(kn)!$ expansion

Question

I am try to expand the factorial $(kn)!$ And got this $$(kn)!=k^{kn}×n!×\prod_{i<k}{(n-\frac{i}{k})}$$

Is my approach right or contain any mistake. I calculated using induction Like n!, (2n)!, (3n)!, And got this general term.

Answer 1

$$(kn)!=\prod_{j=0}^{kn-1}\left(kn-j\right)=k^{kn}\prod_{j=0}^{kn-1}\left(n-\frac{j}k\right)=k^{kn}n!\prod_{0\lt j\lt kn,\,k\nmid j}\left(n-\frac{j}k\right)$$