Question

$20$ persons are setting in a particular arrangement around a circular table. Three persons are to be selected for leaders. The number of ways of selection of three persons such that no two were sitting adjacent to each other is

• A. 600
• B. 900
• C. 800
• D. None of these

Answer 1

Answer: (C)

Total ways of selection without restriction $={ }^{20} C_{3}$
Number of ways of selection when two are adjacent
$=20 \times{ }^{16} C_{1}$
Number of ways of selection when all the three are adjacent $=20$.
Required number of ways $={ }^{20} C_{3}-20 \times 16-20$
$=\frac{20 \times 19 \times 18}{6}-20 \times 16-20=20[57-16-1] $
$=20 \times 40=800$