Zero Factorial

Date: 05/24/99 at 19:10:23
From: Aubree 
Subject: 0!

My question is this:  Why does 0! = 1?

I have no idea about this one. My Algebra II teacher admits that he 
doesn't know. I have also asked the two physics teachers. Everyone 
tells me 'because it does'. I don't like having to accept this logic 
because I am the kind of person who usually needs a reason for things.  
I would really appreciate any help you could provide. Thank you.

Date: 05/24/99 at 23:49:17
From: Doctor Tom
Subject: Re: 0!

"Because it does" isn't a bad answer. Actually, what happens is this - 
if you define it to be anything else, then lots of other formulas in 
math don't work when you plug zero into them.

Here's a simple example: How many ways can you choose k things out of 
a set of n things? For example, how many ways can you choose 2 things 
out of a set of 4? The answer is 6. If the 4 things are A, B, C, and 
D, here they are:

AB, AC, AD, BC, BD, CD

In general, the formula for "how many ways can you choose k things 
from a set of n?" is given by:

  n!
------
k!(n-k)!

In our example, n = 4; k = 2:

  4!         24
-------  =  ----- = 6.
2!(4-2)!     2(2)

How many ways can you choose zero things? Exactly one way, right? Just 
take none of them - anything else you do is wrong. So if k = 0, the 
formula should give 1, right?

   n!           1
--------- =   ----- = 1
0!(n-0)!        0!

If 0! is anything but 1, the formula doesn't work.

There are LOTS of examples like this, but most of them require a lot 
of math.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/