There are 15 points in a plane, out of which m points are collinear and no other three points are collinear. If the total number of triangles that can be formed by joining any three of these points is 445 , then the value of m is

Step-by-step solution

The total number of ways to choose 3 points from 15 points is ¹⁵C₃ . Since m points are collinear, the selection of any 3 points from these m points will not form a triangle. The number of such invalid selections is ^ m C₃ . The total number of triangles formed is given by: ¹⁵C₃ - ^ m C₃ = 445 Calculate ¹⁵C₃ : ¹⁵C₃ = 15 14 13 3 2 1 = 455 Substitute this into the equation: 455 - ^ m C₃ = 445 ^ m C₃ = 10 We know that ⁵C₃ = 10 . Therefore, m = 5 . Answer: 5