Problem with Primes

[dochelay]

Show that if p is prime and 0 < k < p, then (p-k)!(k-1)! = \(\displaystyle (-1)^k\) (mod p)

I'm guessing this has something to do with Wilson's Theorem, but I'm having trouble manipulating it.

 

[daon]

(p-1)! = (p-1)(p-2)...(p-k+1)(p-k)!

Taken mod p, the rhs is equal to (0-1)(0-2)...(0-k+1) = (-1)(-2)(-3)...(-k+1)(p-k)!

Try to figure it out from here...