# A problem on the friends and strangers theorem

There is a group of 20 people. Each pair of people are either friends or strangers, and each person finds exactly 6 strangers in the group. If all possible committees of 3 are formed from the group, what is the total # of committees consisting of all friends or all strangers ?

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Are any two people either friends or strangers, but not both? And is the stranger relation commutative? – Harald Hanche-Olsen Sep 17 '12 at 15:06
Question edited to clarify this point. – true blue anil Sep 17 '12 at 15:13

If there are $N$ 3-groups containing both friends and strangers, the 3-group and member having both friend and stranger can be picked in $2N$ different ways.
On the other hand, if we pick any one of the 20 persons, we can pick one friend and one stranger in $6\times13$ different ways, and the 3-group and member pairs can all be produced in this manner. Hence, $2N=20\times6\times13$ which makes $N=780$.
The total number of ways to pick 3-groups of any kind is ${20\choose3}=1140$. Hence, the number of ways to pick a 3-group with either all friends or all strangers is $1170-780=360$.