X persons stand on the circumference of a circle at distinct points.

Avinasht123
I am not expert, I can try to explain

Let there be \(n\) person standing in a circle. Then we have to form teams with 2 people each, such that these 2 people selected in a team should not be adjacent to each other in the circle.

For first person , we can \(n\) choices, we can select anybody out of \(n\) persons.

For second person, we have \(n-3\) [\(n- 1\)(person already selected)\(-1\)(Person to the right of already selected person)\(-1\)(Person to the left of already selected person)]

Total number of such 2 person team:\(\frac{n(n-3)}{2}\)

Division by \(2\) is required as expression \(n(n-3)\) counts AB,BA as two pairs , whereas they are only one pair

Now Question says each team formed should sing a 2 minute song.

Total time spent by \(\frac{n(n-3)}{2}\) teams singing=\(2* \frac{n(n-3)}{2}\)minutes

Now in question , it is given that \(n(n-3)=28\)
This leads to \(n(n-3)=28\)
\(n^2-3n-28=0\)
\(n^2-7n+4n-28=0\)
\(n(n-7)+4(n-7)=0\)
\((n+4)(n-7)=0\)
\(n=7\) neglecting \(n=-4\)