20 persons are sitting in a particular arrangement around a circular table.Number of ways of selection of three persons from them such that no two were sitting adjacent to each other is
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Solution
The total number of ways of selection without restriction is 20C3.
The number of ways of selection 3 persons when only two are adjacent. Two adjacent persons can be selected as {(1,2),(2,3),...(20,1)}, i.e. 20 ways.
Then the remaining one person can be selected as 16C1 ways. (∵ for eg, if we selected {(3,4)}, 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 {(1,2,3),(2,3,4),...(20,1,2)}, i.e. 20 ways)
∴The required number of ways is =20C3−20×16−20 =20×19×186−20×16−20 =20[57−16−1]=20×40 =800