Congruence in rings Let $R$ be a commutative (and probably unitary, if you like) ring and $p$ a prime number. If $x_1,\ldots,x_n\in R$ are elements of $R$, then we have $(x_1+\cdots+x_n)^p\equiv x_1^p+\cdots+x_n^p$ mod $pR$. Why is this true? I tried to show that in $R/pR$ their congruence classes are equal, but without sucess.

HINT:
When $n=2$ the question reduces immediately to checking that the binomial coefficients $\binom pk$ are multiples of $p$ for $k\in\{1,...,p-1\}$.
Then use induction on $n$.
