Question

if three persons are selected at random from n persons sitting at a round table then the probability that no two of them are adjacent to each other are

Answer 1

There are (nC3)ways to choose 3 people sitting in three consecutive seats.

There are n ways to choose 3 people where all 3 are sitting together in the round table.

Thus we have 7⋅n choices that do not qualify the hypothesis. Excluding these from (nC3) we get the number of ways to choose 3 people out of 10 in a round table with no two persons being adjacent to each other as (nC3)−7n