Complementary factorial congruence ($h!k!\equiv (-1{)}^{h+1}(\mathrm{mod}p)$)

For a prime number $p$ and nonnegative integers $h+k=p-1$, Wilson theorem gives $(-1{)}^{k}h!k!\equiv -1(\mathrm{mod}p)$ by replacing the last $k$ factors of $(p-1)!$ with $-1,-2,\dots ,-k$. For odd $p$, $h,k$ have the same parity; for $p=2$, both signs have the same residue class. Thus $h!k!+(-1{)}^h\equiv 0(\mathrm{mod}p)$, including the endpoint cases $0!=1$.