Question

$ABCD$ is a convex quadrilateral. $3,4,5$ and $6$ points are marked on the sides $AB , BC , CD$ and DA respectively. Find the number of triangles with vertices on different sides.

• A. 215
• B. 342
• C. 225
• D. 424

Answer 1

Answer: (B)

The number of triangles with vertices on sides
$AB , BC , CD ={ }^{3} C _{1} \times{ }^{4} C _{1} \times{ }^{5} C _{1}$
Similarly for other cases
$\therefore $ The total number of triangles
$={ }^{3} C _{1} \times{ }^{4} C _{1} \times{ }^{5} C _{1}+{ }^{3} C _{1} \times{ }^{4} C _{1} \times{ }^{6} C _{1}$
$+{ }^{3} C _{1} \times{ }^{5} C _{1} \times{ }^{6} C _{1} +{ }^{4} C _{1} \times{ }^{5} C _{1} \times{ }^{6} C _{1}=342$