How many "odd committees" are there?

Question

There are 30 councilors in a council. Any two councilors are either friends or political opponent of each other. Each councilor has six political opponents. For a committee of three councilors, if any two of them are friends or any two of them are political opponents of each other, then the committee is called an "odd" committee. How many odd committees are there? (24th U.S.S.R. Math Olympiad)

Answer 1

• let x denote the number of "odd" committees
• let y denote the number of other committees

then $x+y=\frac{30!}{3!27!}=4060$ we have $4060$ selections of $3$ councilors from council for a committee

consider a councilor if we want to select two other members in which the other two members are both his friends or both his political opponents we have $\frac{6!}{2!4!}+ \frac{23!}{2!21!}=268 $ committees.

now consider each councilor so we have $30\times 28=8040 $ committees, but each odd committee mentioned three times, while others mentioned only once each.

so we have $3x+y=8040$, now with solving $x+y=4060$ and $3x+y=8040$ we find $x=1990$