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If I was in a room with 4 other people and wanted to shake everyone's hand, four handshakes would happen. It gets trickier though if I wanted to know how many handshakes there would be if everyone wanted to shake everyone else's hand once. By using diagrams to find out how many handshakes would happen for different numbers of people, a pattern might be found…

At the top of my diagram I had colored five colored dots represent each person in the room. The first person was the red dot. The first person can't shake hands with himself but can shake hands with everyone else. So under the red dot I put a blue dot, a green dot, a yellow dot, and a purple dot. The second person can't shake hands with themselves and already shook hands with the first person. Under the blue dot I put a green dot, a yellow dot, and a purple dot. The third person can't shake hands with themselves and they already shook hands with the first two people. I put a yellow dot and a purple dot under the green dot. The fourth person can't shake hands with themselves and they already shook hands with the first three people, so I only put a purple dot under the yellow dot. Now the fifth person shook hands with everyone, so I didn't put any dots under the purple dot. I counted the dots in each column under the 5 dots on top and added them together to get a total of 10 handshakes. This is what my diagram looked like.

After I made this diagram I noticed a pattern. When there were five people in the room I added up all the numbers under 5, so it was 4+3+2+1. When there were six people in the room I added up all the numbers less than 6, so it was 5+4+3+2+1. ...

Download paper After I made this diagram I noticed a pattern. When there were five people in the room I added up all the numbers under 5, so it was 4+3+2+1. When there were six people in the room I added up all the numbers less than 6, so it was 5+4+3+2+1. ...

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