# simplify (factorial expression)

• Nov 30th 2011, 09:50 AM
livinggourmand
simplify (factorial expression)
x! - [(x-1)!+(x-2)!+(x-3)!...............2!+1!]
• Dec 1st 2011, 07:14 AM
Annatala
Re: simplify (factorial expression)
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.
• Dec 1st 2011, 07:34 AM
Plato
Re: simplify (factorial expression)
Quote:

Originally Posted by Annatala
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.

$5! - \left( {\sum\limits_{k = 1}^4 {(5 - k)!} } \right) = 87$.

If $E_N=N! - \left( {\sum\limits_{k = 1}^{N-1} {(N - k)!} } \right)$ for $N\ge 2$ then we get $1,~3,~15,~87,~567$ for the first five of those.

I have not seen a pattern yet.
• Dec 1st 2011, 10:48 AM
emakarov
Re: simplify (factorial expression)
• Dec 1st 2011, 10:53 AM
Plato
Re: simplify (factorial expression)
Quote:

Originally Posted by emakarov

Also looking at this sequence site we see that this does not to yield much.
• Dec 1st 2011, 05:03 PM
livinggourmand
Re: simplify (factorial expression)
do v have any other expression for sum of factorial ??

may be we can find a range (in terms of n) in which the answer would lie..

I ll try this now...
• Dec 1st 2011, 05:56 PM
Plato
Re: simplify (factorial expression)
Quote:

Originally Posted by livinggourmand
do v have any other expression for sum of factorial ??
may be we can find a range (in terms of n) in which the answer would lie..I ll try this now...

I have no idea what that post says.
If you follow the link in post #5, it is clear that the folks at ATT labs do not know of any simplification of this sequence.
• Dec 2nd 2011, 09:54 AM
Annatala
Re: simplify (factorial expression)
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.