Euclid theory problem on number theory for p divides the binomial coefficient.

Question

let p be a prime and let n be any integer satisfying 1<= n <= p-1. Prove that p divides the binomial coefficient (p,n) = p!/[(p-n)!n!]

i know that p|p! but p does not divides 1/[(p-n)!n!] since 1/[(p-n)!n!] < p.

Answer 1

Note that if $n<p$, then $p$ and $n$ are coprime as otherwise would mean that $p$ divides $n$, which is impossible as $p>n$. Next look at the denumerator of $\binom{p}{n}$. It is $n!(p-n)!$. The question you have to ask yourself is: does $p$ divide that.