Question

$20$ persons are sitting in a particular arrangement around a circular table. The number of ways of selection of three persons from them such that no two were sitting adjacent to each other is

Answer 1

The total number of ways of selection without restriction
$={}^{20}{C}_3$
The number of ways of selecting $3$ persons when only two are adjacent.
Two adjacent persons can be selected as $\left\{\right(1,2),(2,3),...(20,1\left)\right\}$, i.e. $20$ ways.


Then the remaining one person can be selected as ${}^{16}{C}_1$ ways.
$(\because$ for eg, if we selected $\left\{\right(3,4\left)\right\}$, then $2$ and $5$ can't be selected$)$


The number of ways of selection when all the three are adjacent is $20$.


Three adjacent persons can be selected as $\left\{\right(1,2,3),(2,3,4),...(20,1,2\left)\right\}$, i.e. $20$ ways$)$

$\therefore$The required number of ways is
$={}^{20}{C}_3-20\times 16-20=\frac{20\times 19\times 18}{6}-20\times 16-20=20[57-16-1]=20\times 40=800$