Question

Three parallel straight lines $L_{1},\, L_{2}$ and $L_{3}$ lie on the the same plane. Consider $5 $ points on $L_{1},\, 7$ points on $L_{2}$ and $9$ points on $L_{3}$. Then the maximum possible number of triangles formed with vertices at these points, is

• A. 1330
• B. 1200
• C. 1201
• D. 129

Answer 1

Answer: (C)

Number of triangles if one vertex at each line is
$={ }^{5} C_{1} \times{ }^{7} C_{1} \times{ }^{9} C_{1}=315$
Number of triangles if two vertices at $L_{1}$ and remaining at either
$L_{2}$ or $L_{3}$ is $={ }^{5} C_{2} \times{ }^{16} C_{1}=160$
Number of triangles if two vertices at $L_{2}$ and remaining at either
$L_{3}$ or $L_{1}$ is $={ }^{7} C_{2} \times{ }^{14} C_{1}=294$
and number of triangles if two vertices at $L_{3}$ and remaining at either $L_{1}$ or $L_{2}$ is:
${ }^{9} C_{2} \times{ }^{12} C_{1}=432$
So, total maximum possible number of triangles
$=1201$