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### Permutations and Combinations

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Date: 03/11/98 at 16:06:18
From: Andrew Demers
Subject: Permutations and Combinations - What's the Difference?

I am a student at Hershey Middle School. Recently I was working with
my math teacher on combinations and permutations. He admitted that
it was not a strong point of his and e were both wondering what the
difference is between the two things.

Andrew Demers
```

```
Date: 03/12/98 at 15:33:01
From: Doctor Daniel
Subject: Re: Permutations and Combinations - What's the Difference?

Hi there,

Good question!  The difference concerns whether the order of the
objects matters, more or less. In a permutation, the order does
matter, while in a combination, all that matters is which set of
objects was chosen.

Here's an example: Suppose you have eight objects, labeled A through
H. Then the number of permutations of 5 of these eight objects is
8x7x6x5x4; there are eight possibilities for the first, seven for the
second, and so on. And the permutation ACHED is different from HEADC.

For combinations, there are far fewer; here we're really asking how
many subsets of 5 objects of the original eight there are. It
shouldn't be too hard to convince yourself that there are 120 times
fewer combinations than permutations. Why?  Well, consider that ALL
permutations that include the letters FACDE are really part of the
same combination, {A,C,D,E,F}. And there are 120 = 5! = 5x4x3x2x1 of
them.

Here are another couple of tidbits: The number of combinations of K
objects chosen from a set of N is equal to N!/K!(N-K)!, while the
number of permutations is just N!/(N-K)!  Also, why is it that there
are exactly the same number of combinations of 5 letters out of 8 as
there are of 3 letters out of 8 (don't just say "the formula tells me
so...")

-Doctor Daniel,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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