Binomial Coefficient of Prime Plus One Modulo Prime

Theorem
Let $p$ be a prime number.

Then:
 * $\displaystyle 2 \le k \le p-1 \implies \binom {p+1} k \equiv 0 \pmod p$

where $\displaystyle \binom {p+1} k$ is a binomial coefficient.

Proof
From Binomial Coefficient of Prime, we have:
 * $\displaystyle \binom p k \equiv 0 \pmod p$

when $1 \le k \le p-1$.

From Pascal's Rule we have:
 * $\displaystyle \binom {p+1} k = \binom p {k - 1} + \binom p k$

The result follows immediately.