$$ \lim_{n \to infty}\frac{q!\left(nq-q\right)!n^{q}}{nq\left(nq-1\right)\left(nq-2\right)...\left(nq-q\right)!}$$ $$= \lim_{n \to infty}\frac{q!n^{q}}{n^{q}q^{q}}$$
Obviously, the $\left(nq-q\right)!$ cancels out but I am perplexed as to how the rest of the denominator becomes raised to the power of q
$$\lim_{n\to\infty}\frac{q!(nq-q)!n^q}{(nq(nq-1)(nq-2)\cdots(nq-q))!}$$ $$= \lim_{n\to\infty}\frac{q!n^q}{nq(nq-1)(nq-2)\cdots(nq-q+1)}$$ Dividing the numerator and denominator by $n^q$ then gives $$= \lim_{n\to\infty}\frac{q!}{q(q-\frac{1}{n}) (q-\frac{2}{n})\cdots(q-\frac{q-1}{n})}$$ $$=\frac{q!}{q^q}\,.$$