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### Handshakes at a Party

```
Date: 6/23/96 at 21:27:42
From: Anonymous
Subject: Handshakes at a Party

If there is a party and every person shakes hands with each other
once, and there are 45 handshakes, how many people are there at the
party?

I don't have a clue how to solve it.
```

```
Date: 6/24/96 at 5:48:58
From: Doctor Anthony
Subject: Re: Handshakes at a Party

If there are n people at the party, then each person will shake hands
with n-1 other people.  So with n people each making (n-1) handshakes,
it appears at first sight that there are n(n-1) handshakes.

However, each handshake will have been counted twice, i.e. A->B and
B->A, so we must divide by 2.

Total number of handshakes = n(n-1)/2

Now we are given that there were 45 handshakes in all, so we must
solve the equation:

n(n-1)/2 = 45

n(n-1) = 90

n^2 - n - 90 = 0

(n-10)(n+9) = 0      From this n = 10 or -9

Clearly the -9 has no meaning in this question, so we conclude that
n = 10

Number at the party = 10

-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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