Induction question help.

Question

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 $n=k+1$.

Answer 1

Hint:

Show that for all $\;1\le k\le p^n-1\;$ , we have

$$p\mid\binom{p^n}k$$

Answer 2

Note that by the binomial theorem $$ (x+y)^{pn} = \sum_{k=0}^{pn}\binom{pn}{k}x^ky^{pn-k} = \sum_{k=0}^{pn}\frac{(pn)!}{k!(pn-k)!}x^ky^{pn-k} = \sum_{k=0}^{pn}\frac{pn\cdot\ldots\cdot(pn-k+1)}{k!}x^ky^{pn-k} = \sum_{k=0}^{pn}p\frac{n\cdot\ldots\cdot(pn-k+1)}{k!}x^ky^{pn-k} = x^{pn}+y^{pn}, $$ since only when $k=0$ or $pn$ the binomial coefficients dissapear.