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

The number of triangles with vertices on different lines =pC1×pC1×pC1=p3 .The number of triangles with 2 vertices on one line and the third vertex on any of the other two lines =3C1{pC2×2pC1}=6pp(p−1)2 ∴ The required number of triangles=p3+3p2(p−1) .