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.