Prove that if p is prime and 1 less than or equal to k less than p then the binomial coefficient p = (p!)/(k!(p-k)!) is divisible by p.
k
This proves that for any $\displaystyle 1\leq k\leq p-1$ we have that $\displaystyle p(p-1)(p-2)...(p-k+1)$ is divisible by $\displaystyle k!$. Now $\displaystyle \gcd(k!,p) = 1$ because $\displaystyle p$ is a prime this means $\displaystyle (p-1)(p-2)...(p-k+1)$ is divisible by $\displaystyle k!$ thus $\displaystyle p(p-1)(p-2)...(p-k+1)$ is divisibly by $\displaystyle pk!$.