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
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