Question

If $PQRS$ is a convex quadrilateral with $3,4,5$ and $6$ points marked on sides $PQ,QR,RS$ 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 $(PQ,OR,RS+OR,RS$, $PS+RS,PS,PQ+PS,PQ,QR)$
$={}^3{C}_1\times {}^4{C}_1\times {}^6{C}_1+{}^4{C}_1\times {}^6{C}_1\times {}^6{C}_1+{}^5{C}_1\times {}^6{C}_1$
$\times {}^3{C}_1+{}^6{C}_1\times {}^3{C}_1\times {}^4{C}_1$
$=60+120+90+72=342$