The straight lines  $I_1,I_2,I_3$  are parallel and lie in the same plane. A total number of m points are taken on $I_1$  n points on  $I_2,k$ points on  ${\mathrm{I}}_3\cdot$ The maximum number of triangles formed with vertices at these points are: 

Total number of points are m + n + k , the ${\mathrm{\Delta }}^{\mathrm{\prime }}s$  formed by these points  ${=}^{m+n+k}{C}_3$ formed by

 these point ${=}^{m+n+k}{C}_3$

Joining 3 points on the same line gives no triangle, such

${\mathrm{\Delta }}^{\mathrm{\prime }}s$   are ${ }^m{C}_3{+}^n{C}_3{+}^k{C}_3$

Required number    ${=}^{m+n\mathit{+}k}{C}_3{-}^m{C}_3{-}^n{C}_3{-}^k{C}_3$