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$N$ persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is $N$?









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Option(B) is correct

Each person will form a pair with all other persons except the two beside him. Hence he will form $(n–3)$ pairs.

If we consider each person, total pairs $=n(n–3)$ but here each pair is counted twice.

Hence actual number of pairs


They will sing for

\(=\dfrac{n(n-3)}{2}\times 2\)


\(\Rightarrow n=7\)

Hence, n= 7 by discarding -ve value of $n$

(1) Comment(s)


The solution can be explained in a different manner as follows: Each person will form a pair with $(N-3)$ persons(as we have to exclude the two person on either side.

So the total no. pairs formed $\dfrac{N \times (N-3)}{2}$ and they will sing for $\dfrac{N(N-3)}{2}\times2$ minutes (since each pair sings for two minutes)

The total singing time is 28 min..

So $N\times (N-3)=28$

which gives $N=\textbf{7}$ Smile