Prove that this expression is an integer

Question

Prove that

$$ \frac{(p - 1)!}{(p - k)! \cdot k!} $$

is an integer if $0 < k < p$ and $p$ is prime.

Answer 1

HINT: Use your previous question and the fact that $$\frac{(p - 1)!}{(p - k)! \cdot k!}=\frac1p\binom{p}k\;.$$