Question

The sides AB, BC & CA of a triangle ABC have 3, 4 & 5 interior points respectively on them. Find the number of triangles that can be constructed using these interior points as vertices.

• A. $144$
• B. $225$
• C. $1260$
• D. $205$

Answer 1

The correct option is D $205$

All the points will form the triangle expects those which are collinear

So, number of triangles ${{}^{12}C}_3$

i.e. from $12$ points $3$ are choosen.

However on each side collinear point shall be there $={}^3{C}_3+{}^4{C}_3+{}^5{C}_3$

$=1+4+10$

$=15$

Total ${{}^{12}C}_3-15=205.$