Question

Out of 12 points, p points are collinear. No three of them are collinear except these points. If number of triangle with vertices at these points is 185 then value of p is equal to:

Answer 1

Answer:

210

Total number of triangles that can be formed with 12 points (if none of them are collinear) = 12C3

(this is because we can select any three points and form the triangle if they are not collinear)

With collinear points, we cannot make any triangle (as they are in straight line).

Here 5 points are collinear. Therefore we need to subtract 5C3 triangles from the above count.

Hence, required number of triangles = 12C3 - 5C3 = 220 - 10 = 210