Question

If PQRS is a convex quadrilateral with 3, 4, 5 and 6 points marked on side 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

The correct option is D $342$

Total number of triangles $=$ Number of triangles with vertices on sides $\left(\text{PQ, QR, RS + QR, RS, PS + RS, PS, PQ + PS, PQ, QR}\right)$

$={}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1+{}^4{C}_1\times {}^5{C}_1\times {}^6{C}_1+{}^5{C}_1\times {}^6{C}_1\times {}^3{C}_1+{}^6{C}_1\times {}^3{C}_1\times {}^4{C}_1$

$=60+120+90+72=342$