Question

The sides $AB,BC$ and $CA$ of $△ABC$ are marked with $3,4$ and $5$ interior points respectively. Number of triangles that can be constructed using these interior points as vertices is

• A. $205$
• B. $150$
• C. $200$
• D. $210$

Answer 1

The correct option is A $205$

We have total $3+4+5=12$ points.
So, number of triangles that can be formed using $12$ such points $=$ Total number of ways of selecting $3$ points $-$ Number of ways $3$ collinear points are selected
$={}^{12}{C}_3-{}^3{C}_3-{}^4{C}_3-{}^5{C}_3=205$