14 persons are seated at a round table. Find the number of ways of selecting two persons out of them who are not seated adjacent to each other.

Explanation

the seating arrangement of given 14 persons at the round table as shown below.

Number of ways of selecting 2 persons out of 14 persons `=""^(14)C_(2)=(14xx13)/(1xx2)=91`.
In the above arrangement two persons sitting adjacent to each other can be selected in 14 ways. (they are `a_(1)a_(2), a_(2) a_(3), ......., a_(13) a_(14),a_(14), a_(1))`
`therefore` The required no. of ways `=91-41 = 77`