Question

Let \( P_1, P_2, \ldots, P_{15} \) be 15 points on a circle. The number of distinct triangles formed by points \( P_i, P_j, P_k \) such that \( i + j + k \ne 15 \), is:

• A. \( 449 \)
• B. \( 419 \)
• C. \( 455 \)
• D. \( 443 \)

Hint:

Triangle Counting on Circle}
Use \( \binom{n}{3} \) to count all triangles from \( n \) points.
Subtract restricted cases to get valid count.
For conditions like \( i + j + k \ne C \), look for symmetry or direct enumeration.

Answer 1

The Correct Option is D

Total number of triangles formed by 15 points: \[ \binom{15}{3} = 455 \] Number of triangles where \( i + j + k = 15 \) is exactly 12 (based on symmetry or enumeration). Hence: \[ \text{Required number} = 455 - 12 = 443 \]