Congruence with binomial

Question

I tried to prove this by induction on $k$. But I did not manage Let $p$ be a prime. For every $k\in\{0,\cdots,p-1\}$, one has

$$\binom{p-1}k\equiv(-1)^k\pmod p.$$ By Wilson theorem, it suffices to prove that

$$(p-1-k)!k!\equiv(-1)^{k-1}\pmod p.$$

So I tried to prove that by induction on $k$.

Answer 1

Use Pascal's identity and Wilson's theorem:

$${n-1\choose k}+{n-1\choose k-1}={n\choose k}$$

Since ${p\choose k}\equiv 0\mod p$ when $1\le k\le p-1$, the result follows.

Answer 2

HINT:

use Pascal's rule and Strong induction