A club of $n$ members is organized into four committees following two rules:
Each member belongs to exactly two committees, and
each pair of committees has exactly 1 member in common.
Find all possible values of $n$.
A club of $n$ members is organized into four committees following two rules:
Each member belongs to exactly two committees, and
each pair of committees has exactly 1 member in common.
Find all possible values of $n$.
If each member must belong to two committees, he or she must be the member in common to those two committees. There are 6 pairs of committees, requiring 6 common memberships and therefore 6 members.
Any more members joining would give his or her two committees two common members.
Therefore, I would say only $n=6$.