Puzzle 11: Quarelling Quartets
The Ruritanian state choir has 120 members. Some pairs of members are
friends. But, a friend of Carla's friend is not necessarily Carla's friend.
In a tradition dating back to a time when the choir was somewhat smaller, every set of four members from the choir are required to prepare a song together.
The choirmaster has noticed that the most quarrelsome quartets are those where two of the members are friends, but no other friendships exist within the quartet. She has also noticed that the present number of quarelsome quartets is as large as it possibly can be. How many quarrelsome quartets are there?