Question

There are $10$ persons sitting around a circular table from which three persons are selected for the board of directors. The number of ways to select them such that no two persons are adjacent to each other is

Answer 1

Let the number of required ways be $r$.
Number of ways to select 3 persons without restrictions = ${}^{10}{C}_3$
Number of ways to select three members together = $10$
Number of ways to select two members together and one member separately = $10{\times }^6{C}_1$
Then,
$r{=}^{10}{C}_3-(10+10{\times }^6{C}_1)$
$r=50$