Why is 0! 1?

Date: 09/14/97 at 23:19:27
From: Kimberly
Subject: Zerofactoro1

My question is: Why is zerofactoro 1?

I've looked in encyclopedia and my math book, but I can't seem to find 
the answer.

Please help me.

Thank you for your time.

Date: 09/15/97 at 14:30:17
From: Doctor Statman
Subject: Re: Zerofactoro1

Hi Kimberly,

Here is a mathematical reason for 0! to be 1.

Let's count how many ways there are to get 3 heads (H) and 
2 tails (T) out of 5 tosses of a coin.  Let's write down all 
the possibilities:

HHHTT  There are 10 sequences of 3 heads and 2 tails.  Instead
HHTTH  of writing them all down, we could have calculated:  
HTTHH  5!/(3!2!) = 120/(6*2) = 10
TTHHH
HHTHT
HTHTH
THTHH
HTHHT
THHTH
THHHT

Let's do the same thing for 4 heads and 1 tail:

HHHHT  There are 5 ways, and this matches 5!/(4!1!) = 120/24 = 5
HHHTH
HHTHH
HTHHH
THHHH

There is only one way to get all 5 heads, and that is HHHHH

Now the formula would say 5!/(5!0!) and if 0! is anything 
other then 1 then this answer would not be equal to 1, as it
should be.

This formula is called a binomial coefficient.  It means that
if you want to choose r objects from n objects, and you don't
care about the order of the r objects, then the number of ways
to do that is n!/r!(n-r)!

Hope this helps - write back if you want to talk about it
some more.

Sincerely,

Doctor Statman

-Doctor Statman,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/