Prove that If $p$ is prime s.t. $0Prove that If $p$ is prime s.t. $0<n\leq p$ , then $p|[{p! \over {(p-n)!(n)!}}]$.
 I know that if $p|q$ , then $q=kp$, for some integer number $k$.
 But I don’t know how to prove that $p$ divided like above.
Is it working to use proof by induction? 

$(p-n)!$ and $n!$ do not contain $p$ as a factor, so neither of them can divide $p$.  However $\binom{p}{n}$ is an integer, containing $p$ as a factor.  So $p$ divides $\binom{p}{n}$.
