Factorial – Shortcut to Calculate Factorial Without Calculator

factorial

I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck whatsoever.

I'm well aware of the fact that there is a way to calculate any number of sigma (summation notation sigma) but haven't figured out anything for factorials yet.
Could you please show me any method that should do the trick.

E.g. factorial of 10! is 3628800 but how do I calculate it without using any sorts of calculator or calculate the numbers from 10 to 1?

Thanks in advance!

Best Answer

Rewriting the factorial as the Gamma function and Stirling's approximation we get what I think is the closest possible approximation that you could do by hand:

$$n! \approx \sqrt{2 \pi n} \cdot \left( \frac{n}{e} \right)^n$$

Where $e = 2.71828\dots$. Unfortunately, this might not be quicker than multiplying all the numbers together by hand, but it's certainly the only shortcut I can think of that could be done by hand.

Related Solutions

[Math] Factorial division using Pascal’s triangle.

You can express those in terms of binomial coefficients and factorials.

$$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$

For example

$$\frac{100!}{60! 5! 2!} = \frac{100!}{60! 40!} \cdot \frac{40!}{5! 35!} \cdot \frac{35!}{2! 33!} \cdot 33! = \binom{100}{60} \cdot \binom{40}{5} \cdot \binom{35}{2} \cdot 33!$$

Or more generally, if $a_1 \ge a_2 \ge \cdots \ge a_k$ and $a_1 + \cdots + a_k \le n$, we have

$$\frac{n!}{a_1! a_2! \ldots a_k!} = \binom{n}{a_1} \cdot \binom{n-a_1}{a_2} \ldots \binom{n-a_1-\cdots-a_{n-1}}{a_n} \cdot (n-a_1-\cdots-a_{n})!$$

And binomial coefficients are easily computed (without division) using Pascal's triangle relation.

[Math] Summing of factorials to produce perfect cubes

EDIT: Wrote a command to do sum of distinct factorials by greedy algorithm; here are the perfect powers I have, not allowing $0!$.

jagy@phobeusjunior:~$ ./greedy_factorial
Sat Dec 13 12:25:29 PST 2014
0
1    1
8    3    2
9    3    2    1
25    4    1
27    4    2    1
32    4    3    2
121    5    1
128    5    3    2
144    5    4
729    6    3    2    1
841    6    5    1
5041    7    1
5184    7    5    4
45369    8    7    3    2    1
46225    8    7    6    5    4    1
363609    9    6    3    2    1
403225    9    8    4    1
3674889   10    8    7    6    3    2    1
1401602635449   15   14   13   12   11   10    9    8    7    3    2    1
Sat Dec 13 12:26:23 PST 2014

Did not get any more cubes yet.

jagy@phobeusjunior:~$ ./greedy_factorial
0   square root    0 = 
1   square root    1 =  1 
8 = 2^3
9   square root    3 = 3
25   square root    5 = 5
27 = 3^3
32 = 2^5
121   square root    11 = 11
128 = 2^7
144   square root    12 = 2^2 3
729   square root    27 = 3^3
841   square root    29 = 29
5041   square root    71 = 71
5184   square root    72 = 2^3 3^2
45369   square root    213 = 3 71
46225   square root    215 = 5 43
363609   square root    603 = 3^2 67
403225   square root    635 = 5 127
3674889   square root    1917 = 3^3 71
1401602635449   square root    1183893 = 3 394631

Allowing $0!$ adds in the square $4,$ I'm not sure i see any other change. Not sure why the program is so very slow. As I said, if all you want is cubes, you can program in a greedy algorithm which, for each cube, will rapidly attempt a decomposition as distinct factorials (with repeat 1 for $0!$ if you wish ) and report success.

ALLOWING 0! = 1  ***+++***
jagy@phobeusjunior:~$ ./greedy_factorial
    Sat Dec 13 11:46:03 PST 2014
    0  eleventh root  0
    1  eleventh root  1
    4  square root  2
    8  cube root  2
    9  square root  3
    25  square root  5
    27  cube root  3
    32  fifth root  2
    121  square root  11
    128  seventh root  2
    144  square root  12
    729  cube root  9
    841  square root  29
    5041  square root  71
    5184  square root  72
    45369  square root  213
    46225  square root  215
    363609  square root  603
    403225  square root  635
    3674889  square root  1917
    1401602635449  square root  1183893
    Sat Dec 13 11:46:36 PST 2014
    jagy@phobeusjunior:~$

Best Answer

Related Solutions

[Math] Factorial division using Pascal’s triangle.

[Math] Summing of factorials to produce perfect cubes

Related Question