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### Shaking Hands - How many were at the party?

```
Date: 6/20/96 at 10:5:1
From: Ang Kaa Cheng
Subject: Men and Women Shaking Hands

A group of people met at a party. Each person shook hands with
everyone else. Mr. Li shook hands with 3 times as many men as women.
Mrs. Li shook hands with 4 times as many men as women. How many men
and women were there at the party?
```

```
Date: 6/20/96 at 11:54:31
From: Doctor Chaos
Subject: Re: Men and Women Shaking Hands

Great problem. The thing to keep in mind is that Mr. Li does not shake
hands with himself, nor Mrs. Li with herself.  Therefore the number of
men should be equal to 3 times the number of women PLUS 1.  The number
of women therefore must be ONE MORE THAN 1/4 the number of men.  Let's
set W = no. of Women and M = no. of Men.  Then we can set up two
equations:

m= 3w+1  and   w=(m/4)+1

From the first equation, we can solve for w to get:
w = (m-1)/3
and substitute this into our second equation to get:
(m-1)/3 = (m/4)+1

Let's simplify the right side by changing the 1 to 4/4 and adding. Now
we have:
(m-1)/3 = (m+4)/4   (you'll want to work out the steps if this is not
obvious).

Now use the cross-products to obtain:
4m-4 = 3m+12

Solve for m and you'll see that there are 16 men.  Substitute back
into the second equation and you'll find the number of women.

You might also want to use this information to determine the TOTAL
number of of handshakes. You can find it by examining Pascal's
Triangle.

I had fun with this one!   Thanks!

-Doctor Chaos,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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