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+kC₃ 

(b) ^m+n+kC₃ – ^mC₃ – ⁿC₃ – ^kC₃ 

(c) ^mC₃ + ⁿC₃ + ^kC₃ 

(d) m+n+k – ^m+n+kC₃

Answer 1

Correct option: (b) ^m+n+kC₃ – ^mC₃ – ⁿC₃ – ^kC₃

Explanation:

Total number of points = m + n + k which give ^m+n+kC₃ number of triangle, 

But m points on I₁ taking 3 point at a time gives ^mC₃ combinations which produces no triangle. Similarly ⁿC₃ and ^kC₃ number of triangles cannot be formed 

∴ number of triangles = ^m+n+kC₃ – (^mC₃ + ⁿC₃ + ^kC₃)