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 $k$.