Remainders of factorials modulo primes

Question

I am currently working on this problem: Prove that if p is prime, then among $1!,2!,\dots,(p-1)!$ one can find at least $\frac{p-1}{2}$ different remainders modulo $p$. I don't really have any clue how to prove it. Can anyone help me?

Answer 1

The conjecture is that the number $N(p)$ of different remainders modulo $p$ among these factorials $1!,\ldots (p-1)!$, satisfies $$ \lim_{p\to \infty} \frac{N(p)}{p}=1-\frac{1}{e}\sim 0.6321205588285576784 $$ So $N(p)\ge \frac{p-1}{2}$ for $p$ large enough. For $p\le 10^7$ this bound is true by a computer verification - see the duplicate here:

among $ 1!,2!,...,p!$ there are at least $ \sqrt{p}$ different residues in modulo $ p$