A dot is marked at each vertex of a triangle \(ABC\) . Then \(2,3\) and \(7\) more dots are marked on the sides \(AB,BC\) and \(CA\) respectively. How many triangles have their vertices at these dots?
You can divide into casework:
(1) One vertex on AB, one vertex on AC, A.
(2) One vertex on BC, one vertex on BA, B
(3) One vertex on CA, one vertex on CB, C
Adding these up, gives you 120 + 80 + 150 = 350.