1. 7 people take part in a panel discussion. Each person is to shake hands with all of the other participants at the beginning of the discussion. How many handshakes take place? List them all.

When two people shake hands, they form a 2 element subset of the set of participants, so there are as many handshakes as there are 2 element subsets of a 7 element set

21 is not that many handshakes. We can list them all without too much trouble. Let the set of participants be {A, B, C, D, E, F, G}. It is very helpful to have a logical method of proceeding. First we consider all of the people with whom person A shakes hands. Then we list all of the people with whom person B shakes hands. When we list all of the people with whom person B shakes hands, we do not have to list the handshake with person A again, because that handshake was already listed when we considered the handshakes with person A. We do this with all of the participants and get the following list of handshakes.

Person A shakes hands with all 6 other participants. When person A shakes hands with person B, person B has shaken hands with person A, so there are only 5 other participants left with whom person B still has to shake hands. At this point, person C will have only 4 people whose hands need to be shaken. If we carry this line of reasoning through to its end, we will have

handshakes.

This illustrates how the number of 2 element subsets are given by triangular numbers.