Indeed we have to use the binomial theorem. By the way, notice that for the binomial theorem to hold in a ring R, it is very important that the ring is commutative.

So (notice you wrote instead of p at the top of the sum):

Notice that in this sum, we get for the term , and for , we get the term . So if all other terms vanish, then we are done.

The ring has characteristic , so any term, which divides, will vanish. The following result holds true:

Lemma: If , then divides the binomial coefficient .

Notice how this lemma implies that every term of the sum above vanishes, except for and .

You might know this lemma from your book, otherwise let me know.