Question

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

• A. 220
• B. 204
• C. 205
• D. 195

Answer 1

The correct option is C 205

We have in all 12 point. Since 3 points are used to form a triangle, therefore the total number of triangles including the triangles formed by collinear points on Ab, BC, and CA is ${}^{12}{C}_3=220$. But this includes the following:
The number of triangles formed by 3 points on
$AB={}^3{C}_3=1$
The number of triangles formed by 4 points on
$BC={}^4{C}_3=4$.
The number of triangles formed by 5 points on
$CA={}^5{C}_3=10$.
Hence, required number of triangles
$=220-(10+4+1)=205$.