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Math Poster: Handshakes

Date: 08/11/97 at 19:50:52
From: Ralph Vitale
Subject: Math poster

I have to do a poster and figure out the question: if there are 15
people in the room and each person shakes hands with every other 
person, how many handshakes will there be?

I don't have an answer, other than that I can start with 15 and add 7.

Thank you.

Date: 08/11/97 at 21:22:01
From: Doctor Scott
Subject: Re: Math poster

Hi Ralph!

Good question. A great way to solve some problems is to look at a much 
simpler example and then see if you can make a conclusion about the 
more complicated problem.  

Working with 15 people is pretty complicated.  What if we tried 
something easier ... maybe two people.  Well, let's see.  If there are 
2 people in the room, there would be one handshake. (Since people 
don't shake hands with themselves.)  Okay, maybe two people was TOO 
simple and example.  Let's try three.

With three people (let's call them A, B, and C), A shakes hands with 
B, A shakes hands with C, and B shakes hands with C.  That's it, 
right?  Everyone has shaken hands at this point. (Notice that we don't 
have to say that C shakes hands with A, because we've already counted 
that handshake).  Making this ordered list helped a lot, I think.  So, 
there are 3 handshakes.  

How about four people?  Well, if we call them A, B, C, and D, we would 
have:  A-B, A-C, A-D, B-C, B-D, and C-D shaking hands.  There are then 
6 handshakes.  Whew.  It might be a good idea to start building a 
little table of our results:

  number of people     number of handshakes
          2                      1
          3                      3
          4                      6

Okay, one more, maybe.  With five people (call then A, B, C, D, and E) 
we get A-B, A-C, A-D, A-E, B-C, B-D, B-E, C-D, C-E, D-E, or 10 
handshakes.  (We can add this to our table)

If you continue like this you can determine the number of shakes with 
15 people.

You might also continue with the table for a while, maybe for 5, 6, 
and 7 people, and then see if you notice a PATTERN in the number of 
handshakes.  There seems to be one forming (I'll let you think about 
what it is).  

This problem-solving technique of looking at simpler cases and then 
building a table of your results is VERY important in mathematics and 
often helps to solve difficult problems.  Good luck!

Let us know if you need any more help with this problem.

-Doctor Scott,  The Math Forum
 Check out our web site!   

Date: 08/12/97 at 02:41:40
From: Doctor Sarah
Subject: Re: Math poster

Hi there -

In addition to Dr. Scott's answer, you might want to look around in 
the Dr. Math archives for answers to similar questions.  Here are some 
good places to start:

   Counting Handshakes   

   Handshakes at a Party   

   Handshake Problem   

-Doctor Sarah,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Permutations and Combinations

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