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How Many Handshakes?

Date: 08/18/2005 at 20:39:33
From: Victor
Subject: How Many Handshakes

Thirty people at a party shook hands with each other.  How many 
handshakes were there altogether?

Before answering this question, draw a diagram and see if you can 
establish a pattern by collecting data in a table for one, two, three, 
four and five people shaking hands.

If there were 300 people at the party, how many handshakes will there 
be altogether?

I'm not sure how to draw a diagram establishing a pattern of the
number of people and the number of handshakes.

Date: 08/19/2005 at 16:01:51
From: Doctor Wilko
Subject: Re: How Many Handshakes

Hi Victor,

Thanks for writing to Dr. Math!

First, when we say handshakes, we'll agree that we mean Adam shaking 
Bob's hand is the same as Bob shaking Adam's hand, i.e., once two 
people shake hands it is considered a hand shake.

Now let's reason through the handshakes, starting with two people and
working our way up, while looking for a pattern.  

If there are two people at a party, they can shake hands once.  There
is no one else left to shake hands with, so there is only one
handshake total.

  2 people, 1 handshake

If there are three people at a party, the first person can shake hands 
with the two other people (two handshakes).  Person two has already 
shaken hands with person one, but he can still shake hands with person 
three (one handshake).  Person three has shaken hands with both of 
them, so the handshakes are finished.  2 + 1 = 3.

  3 people, 3 handshakes

If there are four people at a party, person one can shake hands with 
three people, person two can shake hands with two new people, and 
person three can shake hands with one person.  3 + 2 + 1 = 6.

  4 people, 6 handshakes

Are you seeing a pattern?

If you have five people, person five shakes four other hands, person 
four shakes three other hands, person three shakes two other hands, 
and person two shakes one hand.  Another way to see it is,

  Person 5  Person 4  Person 3  Person 2

     4    +    3    +    2    +    1    =   10 handshakes total  

People at Party               Number of Handshakes

      2                                           1
      3                      1 + 2              = 3
      4                      1 + 2 + 3          = 6
      5                      1 + 2 + 3 + 4      = 10
      6                      1 + 2 + 3 + 4 + 5  = 15
      .                                             n(n-1)
      n                      1 + 2 + ...+ (n-1) =  --------


It turns out this is just the formula for adding up an arithmetic 
sequence where you know how many terms you have total, and you know 
the first and last terms of the sequence.  See this link for more on

  Adding Arithmetic Sequences 

So, to find how many handshakes there are at a party of 30 people, you 
could add up all the numbers from 1 to 29 or use the formula,

      ----------  = 435 handshakes

Can you figure out how many handshakes there are at a party of 300 

Here's another link from our archives:

  Handshakes at a Party 

Does this help?  Please write back if you have any further questions.

- Doctor Wilko, The Math Forum 

Date: 08/19/2005 at 23:49:46
From: Victor
Subject: Thank you (How Many Handshakes)

Thank you very much Dr. Wilko!
Associated Topics:
High School Permutations and Combinations

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