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

Options

A. 5
B. 7
C. 9
D. none of these

Answer

To solve the problem, we need to determine the number of persons \( N \) standing on the circumference of a circle, given that each possible pair of persons (not standing next to each other) sings a two-minute song, and the total time taken for singing is 28 minutes.

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each person can form a pair with every other person except for the two adjacent persons. Therefore, each person can form pairs with \( N - 3 \) other persons (since they cannot pair with themselves or the two adjacent persons).

2. **Calculating Total Pairs**:
Since there are \( N \) persons, the total number of pairs formed is:
\[ \text{Total Pairs} = N \times (N - 3) \]
This is because each of the \( N \) persons can form \( N - 3 \) pairs.

3. **Total Time Calculation**:
Each pair sings a song for 2 minutes. Therefore, the total time taken for all pairs to sing is:
\[ \text{Total Time} = \text{Total Pairs} \times 2 \]
We know that the total time taken is 28 minutes, so we can set up the equation:
\[ N \times (N - 3) \times 2 = 28 \]

4. **Simplifying the Equation**:
Dividing both sides by 2 gives:
\[ N \times (N - 3) = 14 \]

5. **Rearranging the Equation**:
Expanding the left side:
\[ N^2 - 3N - 14 = 0 \]

6. **Factoring the Quadratic Equation**:
We need to factor the quadratic equation \( N^2 - 3N - 14 = 0 \). We look for two numbers that multiply to -14 and add to -3. The numbers -7 and 4 work:
\[ (N - 7)(N + 4) = 0 \]

7. **Finding the Values of N**:
Setting each factor to zero gives us:
\[ N - 7 = 0 \quad \Rightarrow \quad N = 7 \]
\[ N + 4 = 0 \quad \Rightarrow \quad N = -4 \]
Since \( N \) must be a positive integer, we discard \( N = -4 \).

8. **Conclusion**:
Therefore, the number of persons \( N \) is:
\[ N = 7 \]

### Final Answer:
The value of \( N \) is \( 7 \).

---