20 people at the reunion, how many handshakes take place?

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At a family reunion, everyone greets each other with a handshake. If there are 20 people at the reunion, how many handshakes take place?

We just learned how to do combinations today. But if 20 is the pool(n) what would be the r? Or maybe it isnt a combination questions, I don't know.
How do you do this?
 
Let's look at a smaller case for simplicity. Say 5 people shaking hands.

The first shakes hands with the other 4 people. The 2nd will shake hands with 3 people because they have already shaken hands with the 1st person. The 3rd will have already shaken hands with the 1st and 2nd, so they only shake with 2 more people, and so on. That's 4 + 3 + 2 + 1 = 10

n(n-1)/2.

What would 20 people shaking hands be then?.
 
so 190 from the formula, I would have never been able to figure that formula out though. How come in the formula u divide by 2 and multiply n?
 
A good way to think of it is to think of an n-sided polygon, which has n vertices.

Now, draw diagonals between the vertices, and

also include two lines connecting a particular vertex to the two

adjacent ones. There are (n-1) lines joining any one

vertex to the other vertices in the polygon. If we use each of

the n vertices, each requires (n-1) lines to join to the other

vertices. There are n(n-1) links. But each of these links has

been produced twice, once from each end, and so the number n(n-1) is

too large by a factor of 2. So, you divide by 2.

See?.
 
a b c d e : a shakes hands with b,c,d,e: thats 4 (a goes away!)

b c d e: b shakes hands with c,d,e: that's 3 (b goes away)

c d e: c shakes hands with d and e: that's 2 (c goes away)

d,e : d shakes hands with e: that's 1

ALL cases work that simple way: kapish?
 
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