Proving divisibility of a factorial

Question

Prove that $(k!)!$ is divisible by $k!^{(k-1)!}$

This sheet was about probability so I tried to describe the two numbers divided by each other in the context of probability but I couldn't. I also tried to solve normally but also failed. Help.

Answer 1

Recall that a multinomial coefficient $\dbinom{n}{a_1,a_2,\dots a_n} \in \mathbb{Z}$, where $\sum a_i = n$.

Here $(k-1)!\cdot k = k!$

Hence $\dbinom{k!}{\underbrace{k,k,\dots k}_{(k-1)! \: times}} = \dfrac{(k!)!}{k!^{(k-1)!}} \in \mathbb{Z}$ as required.