# Application of cyclotomic polynomial

1. Nov 30, 2009

### math_grl

1. The problem statement, all variables and given/known data

Show that $${p \choose k} = \sum^{k+1}_{i=1} {p-i \choose p-k-1}$$ where $$\forall k < p \in \mathbb{Z}$$ and $$p$$ a prime.

2. Relevant equations

This is part (b) to a problem. Part (a) is showing that $$1 + x + x^2 + \cdots + x^{p-1}$$ is irreducible in $$\mathbb{Q}[x]$$.

3. The attempt at a solution

I don't know how to apply part (a) to part (b)...Worked out the equality algebraically, then could probably do it by finite induction but think there has to be a shorter easier way that's supposed to follow from the previous part. Help.