Babbage's Congruence

Theorem
Let $p$ be a prime number such that $p \ge 3$.

Let $a, b \in \Z_{\ne 0}$ be non-zero integers.

Then:
 * $\dbinom {a p} {b p} \equiv \dbinom a b \pmod {p^2}$

where $\dbinom a b$ denotes a binomial coefficient.

Also see

 * Wolstenholme's Theorem