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Question

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
= 20C320×1620
=20×19×18620×1620
=20[57161]=20×40
=800

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