Binomial Coefficient of Prime Plus One Modulo Prime

Theorem
Let $p$ be a prime number.

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

where $\dbinom {p + 1} k$ denotes a binomial coefficient.

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

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

From Pascal's Rule:
 * $\dbinom {p + 1} k = \dbinom p {k - 1} + \dbinom p k$

The result follows immediately.