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### Handshake Problem

```
Date: 05/07/97 at 12:44:02
From: Bushnell-Prairie City Elem. School
Subject: Handshake problem

Dr. Math,

Our 5th grade math class was learning to solve story problems by
looking for a pattern and setting up a chart. Most of the problems
were of the nature: If you had 8 people in a group and each one had
to shake everyone else's hand one, how many handshakes would take
place?

After working through several problems, one boy in the room said "I
can get it faster without making a chart. If you multiply whatever
the number is by one less, then divide by 2, you get the right answer
each time."

Will this always work?  Why?

Mitchell C. and Mrs. Holland
```

```
Date: 05/07/97 at 16:46:32
From: Doctor Anthony
Subject: Re: Handshake problem

Yes, it will always work.  The boy concerned was very observant.  A
good way to model the situation is to think of an n-sided polygon,
which has n vertices.

Now consider how many diagonals you can draw between the vertices, and
also include two lines connecting a particular vertex to the two
adjacent ones. It is clear that there are (n-1) lines joining any one
vertex to the other vertices in the polygon. If we consider each of
the n vertices, each requires (n-1) lines to join to the other
vertices. There are thus n(n-1) links. But each of these links has
been produced twice, once from each end, and so the number n(n-1) is
too large by a factor of 2.

Hence total number of connections joining every vertex to every other
one is:

n(n-1)
------
2

This is the result that the boy discovered by observation.

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

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