First, when we say handshakes, lets assume Kate has a friend Adam. We'll agree that we mean Kate shaking Adam's hand is the same as Kate 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.

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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

 

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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:

 

Adding Arithmetic Sequences 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,

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