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### Zero Factorial

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

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

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.
```

```
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/
```
Associated Topics:
Middle School Factorials

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