On the sides AB, BC, CA of a triangle ABC, 3,4,5 distinct points (excluding vertices A, B, C ) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are :

Correct answer: (B) 205. Required number of triangles =^12 C_3- (^3 C_3+^4 C_3+^5 C_3 )=205

Q56 / 90 · JEE Main 2013 (23 Apr Online) · Mathematics

On the sides $\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ of a $\triangle \mathrm{ABC}, 3,4,5$ distinct points (excluding vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ ) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are :

Required number of triangles $$ ={ }^{12} \mathrm{C}_3-\left({ }^3 \mathrm{C}_3+{ }^4 \mathrm{C}_3+{ }^5 \mathrm{C}_3\right)=205 $$

Hence, the correct answer is option: 'B'.