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 verticles on different sides is?

• A. $220$
• B. $270$
• C. $342$
• D. $282$

Answer 1

Answer: (C)

$342$

Total number of triangles $=$ Number of triangles with vertices on sides PQ, QR and RS $+$ Number of triangles with vertices on sides QR, RS and PS $+$ Number of triangles with vertices on sides RS, PS and PQ $+$ Number of triangles with vertices on sides PS, PQ and QR
$={}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1+{}^4{C}_1\times {}^5{C}_1\times {}^6{C}_1+{}^5{C}_1\times {}^6{C}_1\times {}^3{C}_1+{}^6{C}_1\times {}^3{C}_1\times {}^4{C}_1$
$=60+120+90+72=342$.