TITLE: Closed form for $(p-n)!\pmod{p}$ where $p$ is prime
QUESTION [7 upvotes]: Does $(p-n)!\pmod{p}$ have a closed form for any $n>2$ when $p$ is prime?

$(p-0)!=0 \pmod{p}$
$(p-1)!=-1\pmod{p}$
$(p-2)!=1\pmod{p}$

REPLY [7 votes]: Since $(p-1)! \equiv -1 \mod p$, $$(p-n)! \equiv  \frac{(p-1)!}{\prod_{j=1}^{n-1} (p-j)}  \equiv (-1)^{n-1} ((n-1)!)^{-1} \mod p$$