Question

Some persons are standing in a circle, all facing the center. Each pair not adjacent sings a 3-minute song, one after another. Total time = 1 hour. How many persons are there?

• A. 5
• B. 7
• C. 9
• D. 8

Hint:

In a circle, each person has 2 neighbors. Subtract n from total pairs to count non-adjacent. Use total time to solve.

Answer 1

The Correct Option is B

Let number of persons = \( n \)
Each pair of non-adjacent persons sings once → total such pairs = \( \binom{n}{2} - n \) Why subtract \( n \)? Because each person has 2 adjacent neighbors in circle So: \[ \text{Total songs} = \binom{n}{2} - n = \frac{n(n - 1)}{2} - n = \frac{n(n - 3)}{2} \] Each song takes 3 minutes, total time = 60 minutes \[ 3 \times \frac{n(n - 3)}{2} = 60 \Rightarrow \frac{n(n - 3)}{2} = 20 \Rightarrow n(n - 3) = 40 \Rightarrow n^2 - 3n - 40 = 0 \Rightarrow n = 8, n = -5 \Rightarrow \boxed{n = 8} \] Wait! But earlier you gave
(b) 7. So mistake here. Try with n = 7: \[ \frac{7(7 - 3)}{2} = \frac{28}{2} = 14 \Rightarrow 14 \text{ songs} \times 3 = 42 \] n = 8: \[ \frac{8(8 - 3)}{2} = \frac{40}{2} = 20 \Rightarrow 20 \times 3 = 60 \Rightarrow \boxed{n = 8} \]