National Student Meeting

Yesterday we got into groups of around four and had to work out a solution to a given question. The three other people in my group were Louise, Caleb and Madi. There were two other groups around us that were working on the same question. The question went something like this "The Prime Minister of New Zealand invites you to represent your school to discuss the future of NCEA. There are 20 people in total going, including yourself. You are instructed to shake everyone's hand when you get there. If everyone were to shake everyone else's hand, how many handshakes have been made in total?". At first we all thought that the answer was going to simple be something like 19x20. 20 representing the number of people in the room and 19 being the number of handshakes everyone is going to give to everyone else, excluding themselves obviously. The answer to that was 380. But then our teacher came around to all of the groups and when he reached us, he told us to use the same idea, but with a smaller number of people. He took our group as an example. If Caleb were to shake everyone else's hand one, that would be 3. If I were to shake everyone else's hand as well, it would end up being only 2, since I've already shaken Caleb's hand. Then Louise would only have one other person's hand to shake then by the time it went to Madi's turn, she's already shaken everyone's hand. So in total, there were 6 hand shakes made in a group of four. That changed our thinking entirely and had to start again from the beginning. In the end I couldn't find a proper way to find the answer, or what the answer even was to be honest, so our teacher got the groups to share their ideas. The other two groups had the same answer but different techniques. The easiest one that I found was simply a diagram that looked something like this:

You yourself would only have to shake 19 hands. Person 1 would only have to shake 18 other hands since they've already shaken yours. Person 2 would only have to shake 17 hands since they've already shaken yourself and person 1's hands; and so on. The answer in conclusion is 180. It was half of our first idea which was 19x20. If we were to do the group of four like that, 3x4, then the answer would also have been half. The teacher said that there was a reason that happened but he wouldn't tell us why and he left it at that.