Committees
Every member of the club is a member of two committees. How many members are there in the club?
In a club there are four committees.
Every member of the club is a member of exactly two committees.
Each pair of committees has exactly one member in common.
How many members are there in the club?
This problem is adapted from the World Mathematics Championships
Drawing the committees and putting the members into them
In the diagram below, the four circles represent the four committees and member A is in two of them.
Image
Adding members B, C and D in an obvious way gives the diagram below.
Image
Now, each committee shares exactly one member with the committees next to it, but not with the one opposite. Adding members E and F gives the diagram below  showing that there are a total of 6 members.
Image
Using a diagram to represent committees and members
In the diagram below, the four circles represent the four committees. The line represents a member.
As each member of the club is in two committees, each line should connect two circles.
Image
As each pair of committees has exactly one member in common, each circle should be joined to each other circle by one line. This is shown in the diagram below, which contains 6 lines, so there are 6 members in the club.
Image
Counting pairs of committees
Labelling the committees A, B, C and D can help us to count the members.
Committee A shares a member with each of the other committees, so contains 3 members.
Additionally, committee B shares a member with committees C and D  which is 2 more members.
Committees C and D also share a member.
Each member is only in two committees, so each of the members counted above must be a different person.
That gives a total of 3 + 2 + 1 = 6 members.
This information is also represented in the twoway table below.
A  B  C  D  
A    Image

Image
Image

Image

B      Image

Image

C        Image

D         
Working out the total number of members by counting the number in each committee
Each committee shares exactly one member with each other committee  so must contain 3 members.
3 $\times$ 4 = 12, so that makes 12 members altogether. But each member is in two committees  so each member has been counted twice.
So there are 12 $\div$ 2 = 6 members in the club.