Question by Sagar: There are 20 people on the board of directors of a publicly listed company. Each pair of people are either fri?
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?

Best answer:

Answer by M3
consider a 20 sided polygon (icosagon) and join all possible pairs.
color enemy pairs as red, and friend pairs as green.
consider each person a vertex, and each triangle a committee.

let there be N bi-colored triangles, then there are 2N vertices from which both colors emerge.

and if we consider any particular vertex, there are 6*13 ways for both colors emerging, ie 20*6*13 in total, so 2N = 20*6*13 , N = 780

the remaining triangles, 20c3 – 780 = 360 must be monochromatic
ie formed of all friends or all enemies

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ans: 360
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