Question

$20$ persons are sitting 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)

$800$

First we will found the number of ways of selecting $3$ leaders and then subtracting cases when they are together selecting $3$ out of $20$

$={}^{20}{C}_3$

Now we need to subtract the case they are together that gives either $2$ person are adjacent which gives $20$ ways.

For exactly $2$ adjacent $3rd$ difference $=20\left(2adjacent\right)\times 16.$

$=320.$

For exactly $3$ we have $20$ ways.

So, total ways $={}^{20}{C}_3-320-20$

$=800$ ways.

Hence, the answer is $800.$