This is a homework question.
I know there exists infinitely many primes. Let $n = p-1$ and so by Wilson's theorem we know there exists atleast one prime $p$ that divides $n! + 1$. I used wolframalpha and checked for a couple of $n = p-1$ values and all show me that there are in fact two distinct primes and one of them is in fact $p$.
How can I use this to conclude there exists a second prime $q$, $q \neq p$ such that $q$ dividies $n!+1$