Why does 0! = 1 ?

Question

I'm talking about (n) factorial, as in (n!), it has something to do with permutations. And what are good math sites that explain permutation?

Answer 1

because (n+1)! = (n+1) n!
Thus in case n = 0
1 = 1! = 1 * 0!
1 = 0!

Answer 2

The other answerers are all partially correct, and there are a lot of reasons to define 0!=1. No one has yet mentioned the combinatorics reason though, which seems to be the specific reason you're looking for. In terms of permutations, n! is the number of permutations of n objects.

For example, consider 3!: There are 3*2*1=6 ways to arrange the elements of {1,2,3}:
(1,2,3), (1,3,2),
(2,1,3), (2,3,1),
(3,1,2), (3,2,1).

So for 0!, how many ways can we arrange no objects? This can be a bit awkward to think about, but there's one way; no arrangement at all. Using the notation from above, we have to arrange elements of { }, and the one way to do it:
( )

One other big reason that 0!=1 that hasn't been mentioned has to do with the binomial theorem. Rather than delve into that here, here's a link (wikipedia, while overused, is helpful):
http://en.wikipedia.org/wiki/Binomial_theorem

A side note: 0!=1 IS a definition, as the first answerer said, but there is also a reason for that definition.

Answer 3

There *is* a "why" -- it's for algebraic continuity.

3! / 3 = 2!
2! / 2 = 1!
So, by extension, 1! / 1 = 0!
Continuing, (-1)! is undefined.

As it turns out, when you get into high math and define factorial as specific values of a function defined as an integral (the gamma function), this definition follows from that definition. Up in those academic altitudes, being able to take the factorial of half-integers is quite useful.

Answer 4

Because when I hit "0" on my calculator and then "X!" it says "1".

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Answer 5

0! = 1 is a definition...there is no "why".

http://mathworld.wolfram.com/

Answer 6

http://en.wikipedia.org/wiki/Empty_product

Answer 7

who cares?????? it wont help you in real life...