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. ...
I used this rule to see how many handshakes there would be with even more people.
By looking at all these numbers I noticed a shortcut. Every time you go to find the number of handshakes for one more person in a room, you just add one less than the total number of people to the previous total number of handshakes. Using this rule to find the number of handshakes for 14 people you can add 13 to 78, which is 91. That's a lot easier than adding 13+12+11+10+9 and so on all over again.
Another pattern I noticed is that the number of handshakes for 5 people was 5 times 2, the number of handshakes for 6 people was 6 times 2.5, the number of handshakes for 7 people was 7 times 3, and the number to be multiplied kept increasing by half. To find out how many handshakes would happen in a room with 100 people, maybe I could use this rule to see what number I should multiply 100 by, instead of adding 100+99+98+97 and so on all the way down to 1. I started another table to see what number I needed to multiply 100 by.
Once I got to 21 I saw from the numbers in the table that you can get the number to multiply by subtracting one from the number of people then dividing by two. For 100 people in the room, then, you can subtract 1, which is 99, then divide by 2, which is 49.5. This means that to see how many handshakes would happen in a room of 100 people you just have to multiply 100 times 49.5. The total number of handshakes is 4950.
Even though patterns make things a lot easier, they aren't good if they aren't accurate. To make sure my rule was right I used a calculator to add 99+98+97+96 and so on all the way down to 1. Sure enough, it added up to 4950.