Problem Analysis: Number of Non-Adjacent Pairs

The problem involves 'n' persons standing on the circumference of a circle. We need to find the number of pairs that are *not* standing next to each other.

• The total number of distinct pairs possible from 'n' persons is given by the combination formula: $\binom{n}{2} = \frac{n(n-1)}{2}$.
• In a circle arrangement of 'n' persons, there are 'n' adjacent pairs (person 1 with person 2, person 2 with person 3, ..., person n with person 1).
• Therefore, the number of non-adjacent pairs is the total pairs minus the adjacent pairs: $ \text{Number of non-adjacent pairs} = \binom{n}{2} - n $.

Calculating Total Singing Time

Each non-adjacent pair sings a song for 2 minutes. The total singing time is given as 40 minutes.

• Equation: (Number of non-adjacent pairs) $\times$ (Song duration) = Total time
• Substituting the values: $ \left( \binom{n}{2} - n \right) \times 2 \text{ minutes} = 40 \text{ minutes} $.

Solving for 'n'

1. Divide both sides by 2: $ \binom{n}{2} - n = \frac{40}{2} $
2. Simplify: $ \frac{n(n-1)}{2} - n = 20 $
3. Multiply the entire equation by 2 to eliminate the fraction: $ n(n-1) - 2n = 40 $
4. Expand and simplify: $ n^2 - n - 2n = 40 $
5. Combine terms: $ n^2 - 3n = 40 $
6. Rearrange into a standard quadratic equation: $ n^2 - 3n - 40 = 0 $
7. Factor the quadratic equation: We look for two numbers that multiply to -40 and add to -3. These numbers are -8 and 5. So, the equation factors as: $ (n-8)(n+5) = 0 $.
8. Solve for 'n': This gives two possible solutions: $ n = 8 $ or $ n = -5 $.
9. Select the valid solution: Since 'n' represents the number of persons, it must be a positive integer. Therefore, $ n = 8 $.

The value of 'n' is 8.