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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) =  --------
2

=====================================================

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
that:

http://mathforum.org/library/drmath/view/55724.html

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,

(30)(29)
----------  = 435 handshakes
2

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

Here's another link from our archives:

Handshakes at a Party
http://mathforum.org/library/drmath/view/56139.html

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

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

```

```
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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