Intuitive guess, based on 5:
expression is 5*4*3*2*1 - 4*3*2*1 - 3*2*1 - 2*1 - 1 = 4*4*3*2*1 - 3*2*1 - 2*1 - 1 = 4*3*2*2*1 etc.
So I think the answer is going to be (x-1)!, though that's not a formal proof.
See also the MathWorld article on the sum of factorials.
Also looking at this sequence site we see that this does not to yield much.
Ah, I figured out the mistake in my quickie guess-calculation.
a * b * c - b * c = (a-1) * b * c is true, but a * b * c - c = (a * (b-1) * c) is false. It should = ((a * b) - 1) * c, which doesn't simplify and gets worse the longer the expression is.