Question

If $P Q R S$ is a convex quadrilateral with $3,4,5$ and $6$ points marked on sides $P Q, Q R, R S$ and $PS$ respectively. Then, the number of triangles with vertices on different sides is

• A. 220
• B. 270
• C. 282
• D. 342

Answer 1

Answer: (D)

Total number of triangles $=$ Number of triangles with vertices on sides $(P Q, O R, R S+O R, R S$, $P S+R S, P S, P Q+P S, P Q, Q R)$
$={ }^{3} C_{1} \times{ }^{4} C_{1} \times{ }^{6} C_{1}+{ }^{4} C_{1} \times{ }^{6} C_{1} \times{ }^{6} C_{1}+{ }^{5} C_{1} \times{ }^{6} C_{1} $
$\times{ }^{3} C_{1}+{ }^{6} C_{1} \times{ }^{3} C_{1} \times{ }^{4} C_{1}$
$=60+120+90+72=342$