Induction question help. Let $x$ and $y$ belong to a commutative ring $R$ with prime characteristic $p$.
Show that, for all positive integers $n$
$$ (( x + y )^p)^n   =   (x^p)^n   + (y^p)^n $$
I hope you can can understand notation.
We have to use induction on $n$.
For $n=1$  $  (x + y)^p   $  = $ x^p $ + $ y^p $
Assume for $n=k$
I have almost done it. I am having trouble with $n=k+1$.

Hint:
Show that for all $\;1\le k\le p^n-1\;$ , we have 
$$p\mid\binom{p^n}k$$
