Question

There are $2$ brothers among a group of $20$ persons. The number of ways the group can be arranged around a circle so that there is exactly one person between the two brothers is

• A. $2\times 17!$
• B. $18!\times 18$
• C. $2\times 18!$
• D. $2\times 17!\times 17$

Answer 1

The correct option is C $2\times 18!$

The person who sits in between the two brothers can be selected in${}^{18}{C}_1=18$ ways.
Consider the two brothers and the person in between the brothers as one unit.
Then the number of distinct units are $20-3+1=18$

Now the brothers can be arranged on either side of the person in $2!$ ways.

Therefore, the total number of ways
$=18(18-1)!\times 2$
$=2\times 18!$