Binomial Coefficient of Prime Plus One Modulo Prime

Theorem
Let $$p$$ be a prime number.

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

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

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

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

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

The result follows immediately.