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 $n=k+1$.

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.
