Three parallel straight lines L1, L2 and L3 lie on the the same plane. Consider 5 points on L1, 7 points on L2 and 9 points on L3. Then the maximum possible number of triangles formed with vertices at these points, is

Question

Three parallel straight lines L1, L2 and L3 lie on the the same plane. Consider 5 points on L1, 7 points on L2 and 9 points on L3. Then the maximum possible number of triangles formed with vertices at these points, is (A) 1330 (B) 1200 (C) 1201 (D) 129

Tardigrade Admin · Accepted Answer

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

Q. Three parallel straight lines L1​,L2​ and L3​ lie on the the same plane. Consider 5 points on L1​,7 points on L2​ and 9 points on L3​. Then the maximum possible number of triangles formed with vertices at these points, is

1330

1200

1201

129

Q. 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