Answer to Question #15026 in Combinatorics | Number Theory for abdel007

Question

There are 20 people on the board of directors of a publicly listed company. Each pair of people are either friends or enemies with each other. Every person has exactly 6 enemies on the board. If every group of 3 directors form a committee, what is the total number of committees that are formed by all friends or all enemies?

Answer 1

At first, let's see the possibilities of one person. A person has 6 enemies and 14 friends.
He can form a committee of other two members by either selecting 2 from 6 enemies or 2 from 14 friends. That can be done in

N = C(6,2) * C(14,2).

Now, all the 20 persons have the possibility of

M = 20 * N = 20 * C(6,2) * C(14,2) = 27300.