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

$=({}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1)+({}^3{C}_1\times {}^4{C}_1\times {}^6{C}_1)+\left({}^3{C}_1{}^5{C}_1{}^6{C}_1\right)+\left({}^4{C}_1{}^5{C}_1{}^6{C}_1\right)$ $=342$