How Many Handshakes?

Date: 01/29/2001 at 17:32:48
From: Kathy
Subject: How many handshakes?

I'm trying to help my daughter figure out how to set up this problem: 

There are 40 people in a room. They shake each other's hands once and 
only once. How many handshakes are there altogether?

We know that that 40 people shake hands, so each person shakes 39 
hands. The first person shakes 39 hands, the second person shakes 38 
hands, the third person shakes 37 hands, and so on. 

How do you put this into a formula to solve the problem?

Date: 01/29/2001 at 18:39:36
From: Doctor Greenie
Subject: Re: How many handshakes?

Hi, Kathy -

This is a great problem for young math students to work on using two 
completely different approaches, so they can see that they get the 
same answer.

You have a good start on the first approach. The total number of 
handshakes is

    39 + 38 + 37 + 36 + ... + 3 + 2 + 1

To find this total without having to add all these numbers together, 
notice the following:

  The average of the first and last numbers is 20: 
    (39+1)/2 = 20.

  The average of the second and next-to-last numbers is 20: 
    (38+2)/2 = 20.

If you think about it, you should be able to see that all the numbers 
can be paired in groups of two whose average is 20. You will have 39 
numbers (1 to 39) whose average is 20 - so what is the sum of all 
those numbers? It is just the number of numbers multiplied by the 
average of the numbers: 39*20 = 780.

Here is the other (more mathematically sophisticated) approach:

Each handshake consists of two people shaking hands. How many ways can 
you choose those two people if there are 40 people in the group? You 
can choose any one of 40 people for the first person involved in the 
handshake; then for each of those 40 choices, you have 39 choices for 
the second person. So you have 40*39 = 1560 ways of choosing the two 
people for the handshake.  

But wait a minute. Person A shaking hands with person B is the same as 
person B shaking hands with person A; this means that in counting the 
1560 handshakes you have actually counted each one twice, so the 
actual number of handshakes is half of 1560, or 780.

Two seemingly completely different ways to look at the problem, and 
they give the same answer!

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/