So we are asked to show that $(p-1)(p-2)\cdots(p-r)\equiv (-1)^{r}r! \pmod{p}$ for $r=1,2,...,p-1$. I worked on it and I want to know if my proof suffices to show what is being asked. I would also appreciate any alternative proofs.
Proof:
We know that $(p-1)\equiv -1\pmod{p},\,(p-2)\equiv -2\pmod{p}, \,\,\ \cdots \,,(p-r)\equiv -r\pmod{p}$. Multiplying all of these together we have $(p-1)(p-2)\cdots(p-r)\equiv (-1)(-2)\cdots(-r)\pmod{p}$ which is the same $(p-1)(p-2)\cdots(p-r)\equiv (-1)^r r!\pmod{p}$ Q.E.D.
Yes? No? Any suggestions. Thank you in advance.