Is there any better way to find n! without just multiplying all the numbers till n?

Question

I was solving some permutation, combination problems where we know that we are to use factorial. So I was thinking if there was any shorter way, maybe a formula, than multiplying all the numbers till $n$ to find $n$!

I know how to add series like $1+2+3+4+…+n$ which is $\frac{n(n+1)}{2}$ but I didn't find much on how to multiply. So... how to do it then?

I do not think so. Any method will comprise of you multiplying something with something else. — Sntn, Feb 10 '16 at 12:24
I mean what if you are asked the answer of "194!" Is there any alternative way to fond out the ans without just multiplying from 1 to 194? — Farhan Fuad, Feb 10 '16 at 12:26
So there is no formula invented like n(n+1)/2 for this problem yet? that's odd — Farhan Fuad, Feb 10 '16 at 12:30
@Farhan there are nice approximations. However, if you want the exact answer, there no neat condensed form like there is for addition. — Ben Grossmann, Feb 10 '16 at 12:32
@Farhan one analog to the $n(n+1)/2$ is that you can simplify $b^nb^{n-1}\cdots b^1$ — Ben Grossmann, Feb 10 '16 at 12:36
http://mathworld.wolfram.com/PochhammerSymbol.html formula (8) gives a 'closed form' for the Pochhammer symbol. — gammatester, Feb 10 '16 at 12:37

gammatester · Answer 1 · 2016-02-10T13:21:17.453

After your editing of the original question:

You will not find an easy formula for the exact evaluation of factorials, except using $n! = \Gamma(n+1)$. If you want to compute faster than just multiply the numbers, use e.g. the prime power table approch.

A basic fast algorithm is described in P.B. Borwein, On the complexity of calculating factorials, Journal of Algorithms, Vol.6, pp. 376-380, 1985. Available as http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P29.pdf

Here the four steps of Borwein's algorithm for $n!$:

Construct the prime table $2 \le p_k \le n\;$ (e.g. using a sieve method)
Compute the exponents of the $p_k$ in $n!\;$ (e.g. Legendre's formula)
Compute $O(\log n)$ numbers $\alpha_i$ (for details see the paper)
Compute $n! = \prod \alpha_i$

For actual implementations see e.g. the mentioned FastFactorial Functions page.

Is there any better way to find n! without just multiplying all the numbers till n?

1 Answers1