Question

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 \( N \)?

• A. 5
• B. 7
• C. 9
• D. None of the above

Hint:

In problems involving combinations, use \( \binom{N}{2} \) to calculate the number of pairs and apply constraints to find \( N \).

Answer 1

The Correct Option is C

The number of pairs of persons that can sing is given by \( \binom{N}{2} - N \), where \( N \) is the number of people. Since each pair sings for 2 minutes, the total singing time is \( 2 \times (\binom{N}{2} - N) = 28 \), solving this gives \( N = 9 \). \[ \boxed{9} \]