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### Factorials

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Date: Tue, 15 Nov 1994 10:16:54 -0500
From: steve barkin
Subject: Chris Grant's factorial issue

Dear Dr. Math

My name is Chris Grant. I am using my teacher's mailbox with his
permission. One day when I was playing on a calculator, I found that 1 and 0
both have a factorial of 1. Can you explain this?

Thank you, Chris Grant.

Steve Barkin
Graham & Parks School
Cambridge, Ma 02139
```

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Date: Mon, 15 Nov 1993 16:30:29 -0500
From: Demetri Bonaros
Subject: Re: Chris Grants factorial issue

Hello, Chris!

Thank you for writing to Dr. Math. Your question is indeed a very good
one, one that puzzles a lot of students when they are first introduced to
the fact that 0!=1. I think there a couple of reasons that explain this
fact; offhand I can only think of one of them, but I'm pretty sure that
it's not the only one.

First, let's see what the binomial coefficients are. We define them as
follows (OK, this is going to be a little hard to do on email, but here
goes, anyway)

/k\
\i/ (the / and\ are supposed to form one big parenthesis) is called "k
choose i". Its value is the answer to the question "in how many ways can
we choose i objects out of a set of k?" The value of "k choose i" is:

k!
__________    (can you prove this? Try induction)
i!(k-i)!

Let's see a few examples: "k choose 1" = in how many ways can you
choose 1 object from a set of k? Well, the answer has to be k, right? If
you have k objects, you have k choices.

"k choose (k-1)" = in how many ways can you
choose (k-1) objects out of a set of k? The answer here is also k. In each
of the ways you are choosing, you are excluding one object (in other words,
you are choosing one object not to choose- does this make sense?) So, this
is the same as "k choose 1".

It turns out that     / k \                  /  k  \
\ i / is equal to      \ k-i / (can you see why?)

Anyway, in this case, "k choose zero" and "k choose k" are

k!                     k!
__________      and    ___________ , respectively.
0!(k-0)!               k!(k-k)!

This would pose a few problems if we defined 0!=0, since the
denominator would be zero in that case. Also, we can figure out without
calculations that the way to choose k objects out of a set of k is only one
(we can pick them all in only one way). Hence, 0! has to be 1, so that the
numerator and denominator of the above fractions will be equal.

I think that 1!=1 follows directly the definition of factorial which is

n!=n(n-1)(n-2)(n-3)...3*2*1

(I suppose I should have put this earlier in the message, but I assumed
that you knew it.)

Anyway, this is one reason we define 0!=1. I guess you could say that
necessity dictates that we do so.

I hope all this makes sense. If not, feel free to write back.

Demetri- Dr, or something...
```
Associated Topics:
Middle School Factorials

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