Elementary Number Theory – Prove $\binom{p-1}{k} \equiv (-1)^k\pmod p$

Question

Prove that if $p$ is an odd prime and $k$ is an integer satisfying $1\leq k \leq p-1$,then the binomial coefficient
$$\binom{p-1}{k} \equiv (-1)^k\pmod p$$

I have tried basic things like expanding the left hand side to $\frac{(p-1)(p-2)………(p-k)}{k!}$ but couldn't get far enough.

Answer 1

Hint: $(p-1)(p-2)\cdots(p-k)\equiv(-1)(-2)\cdots(-k)$ because $p\equiv 0$.