There are three coplanar parallel lines. If any n points are taken on each of the lines, the maximum number of triangles with vertices at these points, is

The maximum number of triangles 

= Number of triangles with vertices on different lines 

+ Number of triangles with 2 vertices on one line and 

third vertex on any one of the remaining two lines

${=}^n{C}_1{\times }^n{C}_1{\times }^n{C}_1{+}^3{C}_1\times \left({ }^n{C}_2{\times }^{2n}{C}_1\right)=3n^2(n-1)+n^3$.