Question

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

• A. $243$
• B. $342$
• C. $432$
• D. $534$

Answer 1

The correct option is B $342$

The number of triangles with vertices on $AB,BC,CD$
$={}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1$

Similarly, for other cases, the total number of triangles
$=({}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1)+({}^3{C}_1\times {}^4{C}_1\times {}^6{C}_1)+({}^3{C}_1\times {}^5{C}_1\times {}^6{C}_1)+({}^4{C}_1\times {}^5{C}_1\times {}^6{C}_1)$
$=342$