Question Detail

To form a triangle, we need to choose 3 non-collinear points.Total number of points = 12.The total number of ways to choose 3 points from 12 is 12C3.12C3 = (12 × 11 × 10) / (3 × 2 × 1) = 2 × 11 × 10 = 220.However, 7 of these points are collinear. If we choose any 3 points from these 7 collinear points, they will not form a triangle.The number of ways to choose 3 points from the 7 collinear points is 7C3.7C3 = (7 × 6 × 5) / (3 × 2 × 1) = 35.The number of actual triangles formed is the total number of ways to choose 3 points minus the number of ways to choose 3 points from the collinear set.Number of triangles = 220 - 35 = 185.