p points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

Step-by-step solution

Total number of points in a plane is 3 p . Maximum number of triangles = ^3 ^p C₃-3 ^p C₃ [Here, we subtract those triangles which points are in a line] aligned & = (3 p) ! (3 p-3) ! 3 ! -3 p ! (p-3) ! 3 ! & = 3 p(3 p-1)(3 p-2) 3 2 - 3 p(p-1)(p-2) 3 2 & = p 2 [9 p^2-9 p+2- (p^2-3 p+2 ) ] & =p^2(4 p-3) aligned