What is the simplest way to show that ${(p-1)! \over (k)!(p-k)!}$ is an integer?

Question

In the proof of $p$ | $\binom{p}{k}$ (p divides $\binom{p}{k}$) where $p$ is prime,

what is the simplest way to show that $${(p-1)! \over (k)!(p-k)!}$$ is an integer?

Answer 1

Multiply the numerator by $p$ to get $\binom{p}{k}$, a natural number and note that the denominator can't have $p$ as its factor.