Show that if p is an odd prime, then$\displaystyle \begin{Bmatrix}p\\ p-1\end{Bmatrix}$is a multiple of p.

I tried to do this using recursion, but I failed to do so because I couldn't complete the

"if hypotheses is true for nth prime then it holds for the (n+1)th prime" step,

since I can't explicitly write down what the next prime is.

I tried to do a combinatorial approach of "distributing p distinct objects into p-1 identical classes", but I couldn't see the connection.

So I would like some hints.

How should I use p's prime-ness in my proof?

Also, is a combinatorial proof possible?