What is a factorial
A factorial is represented by the following pattern:
1!=1
2!=2*1
3!=3*2*1
More generally the algorithm is as follows:
n!=(n-1)(n-2)(n-3)... for n=1,2,3....
Notice that I purposely skipped 0! because it might not make sense yet since 0!=1. Moreover all fractions and all negative numbers were not included in the definition of n! This definition is very limiting so there is a need for a better definition of a factorial for such applications.
A more versatile definition of a factorial
This definition extends what you can do significantly and now the 0! can be shown as follows:
There is a lot more you can do with this formula for exmaple now, it is possible to move away from integers and begin to use fractions of the factorial.
1/2! is solved as follows:
The n can essentially be anything just for show lets plug into mathematica 1!, 2!, and 3!
1!
2!
3!
Honestly I really did not want to do this by hand because it was too much work hopefully you all prove this to yourselves by hand if you don't believe me.
Must be pretty nice for probability