Why does 0! = 1? [duplicate]

Solution 1:

Mostly it is based on convention, when one wants to define the quantity $\binom{n}{0} = \frac{n!}{n! 0!}$ for example. An intuitive way to look at it is $n!$ counts the number of ways to arrange $n$ distinct objects in a line, and there is only one way to arrange nothing.

Solution 2:

In a combinatorial sense, $n!$ refers to the number of ways of permuting $n$ objects. There is exactly one way to permute 0 objects, that is doing nothing, so $0!=1$.

There are plenty of resources that already answer this question. Also see:

http://mathforum.org/library/drmath/view/57905.html

http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one

http://en.wikipedia.org/wiki/Factorial#Definition