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### What Is N Factorial Used For?

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Date: 08/29/2001 at 18:02:22
From: Stuart Reed
Subject: What is n factorial used for?

Can you please tell me what n! or n factorial is used for. I know what
it is (5! is 120, because 1*2*3*4*5 = 120), but not what it is used
for. I have heard that it is used in probability, but what for, and
how? How does it work, and what are its uses?
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Date: 08/30/2001 at 10:49:46
From: Doctor Ian
Subject: Re: What is n factorial used for?

Hi Stuart,

The factorial function is useful in computing the number of
combinations or permutations that can be constructed from a set of

Permutations and Combinations
http://mathforum.org/dr.math/faq/faq.comb.perm.html

Very briefly, here is the kind of problem that would give rise to the
factorial function.

Suppose you have three people (Alan, Bob, Cindy), and you want to know
how many ways they can stand in line. For a small number like 3, the
answer is easy to construct by trial and error:

Alan, Bob, Cindy
Alan, Cindy, Bob
Bob, Alan, Cindy
Bob, Cindy, Alan
Cindy, Alan, Bob
Cindy, Bob, Alan

But for larger numbers of people, it becomes very difficult to avoid
duplicating some sequences, or skipping some by accident.

With a little thought, we can see that if we have N people, then there
are N choices for who might go into the first position:

1. Alan
2. Bob
3. Cindy
4. Diane

Having made that first choice, there are N-1 choices for who might go
in the second position, for _each_ of the original choices:

1. Alan   1. Bob
2. Cindy
3. Diane

2. Bob    1. Alan
2. Cindy
3. Diane

3. Cindy  1. Alan
2. Bob
3. Diane

4. Diane  1. Alan
2. Bob
3. Cindy

And there are N-2 choices for who might go into the third position,
again for _each_ of the previous sets of choices:

1. Alan   1. Bob      1. Cindy
2. Diane

2. Cindy    1. Bob
2. Diane

3. Diane    1. Bob
2. Cindy

2. Bob    1. Alan     etc.
2. Cindy
3. Diane

3. Cindy  1. Alan
2. Bob
3. Diane

4. Diane  1. Alan
2. Bob
3. Cindy

I'll leave the rest of the chart for you to fill in, if you wish.

Note that the final number of possible orderings of N objects will be

ways to
make the
next choice
|
v
N * (N-1) * (N-2) * ... * 1

^                         ^
|                         |
ways to                   ways to
make the                  make the
first choice              final choice

which is exactly the factorial function.

When dealing with combinations and permutations (and, by extension,
when dealing with probabilities), this kind of reasoning comes up over
and over again, which is why the factorial function is so important.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Permutations and Combinations
High School Probability
Middle School Factorials
Middle School Probability

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