Question

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

• A. $205$
• B. $220$
• C. $225$
• D. $230$

Answer 1

The correct option is A $205$

Firstly there are 12 points out of which 3, 4 and 5 are collinear. To form a triangle 3 non-collinear points are required.

Total number of ways of choosing 3 points from 12 points ${=}^{12}{C}_3$

Now we need to subtract 3 points which are collinear

$\therefore$ Total number of triangles that can be formed ${=}^{12}{C}_3{-}^3{C}_3{-}^4{C}_3{-}^5{C}_3=205$