What is the number of triangles that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line?

Step-by-step solution

To form a triangle, we need 3 points. 12 points are given. So, ¹² C ₃ triangles can be formed. But, given that 7 points are on a straight line. selecting 3 points from this set will not form a triangle. So, number of triangles formed = ¹² C ₃- ^7 C ₃ aligned & = 12! 3!9! - 7! 3!4! & = 12 11 10 3 2 1 - 7 6 5 3 2 1 =220-35=185 aligned