Question

The straight lines I₁, I₂, I₃ are parallel and lie in the same plane. A total number of m points are taken on I₁ ; n points on I₂ , k points on I₃. The maximum number of triangles formed with vertices at these points are –

• A. ^{m + n + k}C₃
• B. ^{m + n +k}C₃ – ^mC₃ – ⁿC₃ – ^kC₃
• C. ^mC₃ + ⁿC₃ + ^kC₃
• D. None of these

Answer 1

Answer: (B)

^{m + n +k}C₃ – ^mC₃ – ⁿC₃ – ^kC₃

Answer 2

Total number of points = m +n + k. Therefore the total number of triangles formed by these points is ^{m + n + k}C₃. But out of these m + n + k points, m points lie on I₁, n points lie on I₂ and k points lie on I₃ and by joining three points on the same line we do not obtain a triangle. Hence the total number of triangles is ^{m + n + k}C₃ – ^mC₃ – ⁿC₃ – ^kC₃.