What is Zero-Factorial?

Simple answer: 0! (read "Zero Factorial") is defined to equal 1.

Involved answer(s):
There are several proofs that have been offered to support this common definition.

Example (1)

If n! is defined as the product of all positive integers from 1 to n, then:
1! = 1*1 = 1
2! = 1*2 = 2
3! = 1*2*3 = 6
4! = 1*2*3*4 = 24
...
n! = 1*2*3*...*(n-2)*(n-1)*n
and so on.
Logically, n! can also be expressed n*(n-1)! .

Therefore, at n=1, using n! = n*(n-1)!
1! = 1*0!
which simplifies to 1 = 0!

Example (2)

The idea of the factorial (in simple terms) is used to compute the number of permutations (combinations) of arranging a set of n numbers.

n:	Number of Permutations (n!):	Visual example:
1	1	{1}
2	2	{1,2}, {2,1}
3	6	{1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}
10	3,628,800	ummm, you get the idea...

Therefore,

0

1

{ }

It can be said that an empty set can only be ordered one way, so 0! = 1.