Question

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

• A. 150
• B. 175
• C. 205
• D. 230

Answer 1

Answer: (C)

205

All these $3+4+5=12$ points lie in a plane.

The number of all possible triangles formed joining these points,taking $3$ at a time ${=}^{12}{C}_3=220$

Number of triangles formed by $3$ points, had they not been in straight line (i.e., on $AB$) ${=}^3{C}_3=1$

Number of triangles formed by $4$ points, had they not been in a straight line (i.e., on $BC$) ${=}^4{C}_3=4$

Number of triangles formed by $5$ points, had they not been in a straight line (i.e., on $CA$) ${=}^5{C}_3=10$

$\therefore$ Required number of triangles $=220-1-4-10=205$