A huge sum of products

[harazi]

Let m,n>2 and let p>2 be a prime divisor of both m and n. Consider all products of the form x_1*x_2*...*x_m, where x_i belong to {1,2,...,n} and their sum is a multiple of p. Prove that the sum of all these products is a multiple of (n/p)^m.

[pluricomplex]

Sorry but if all the xi are distinc?

[harazi]

Maybe I didn't express what I thought:
Prove that (n/p)^m is a divisor of \sum x_1*x_2*...*x_m, where the sumation is taken for all systems (x_1,...,x_m) of numbers from {1,2,...,n} having their sum a multiple of p. So, in this sum omly systems which have the sum of their components divisible by p appear.