In the Hot Air Balloon Club each member is on two committees and any two committees have exactly one member in common. There are 5 committees altogether. How many members are in the club?

## Solution

Draw a figure! The statement of this problem conceals its geometric nature. We represent the committees by points. We are told there are five of them (see the figure, part A). We are also told that any two committees have exactly one member in common. If we represent the members by lines, then every pair of committees (that is, pair of points) has a member in common (that is, a line connecting them). The figure, part B, gives the solution. Five points determine ten lines. That is how many members are in the club. It turns out that each committee has 4 members.

By the way, if you haven't already run across it, take a look at #1040.11.