Let $p$ be prime. Show that
$$ (p− 1)(p− 2) \ldots (p − n) \equiv (−1)^{n} \; n! \pmod p $$
where $ 1 ≤ n < p$.
We proceed as follows
$$ \begin{align} (p-1) \times (p-2) \times \ldots \times (p-n) & \equiv \underbrace{(-1) \times (-2) \times \ldots (-n)}_{n \text{ multiplicands}} \pmod p \\ \\ & \equiv (-1)^n \times 1 \times 2 \ldots \times n \pmod p \\ \\ & \equiv (-1)^n \; n! \pmod p \end{align} $$