Question

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

• A. 60
• B. 205
• C. 115
• D. 405

Answer 1

The correct option is B 205

No. of ways $={}^3{C}_1\times {}^4{C}_1\times {}^5{C}_1+{}^3{C}_2({}^4{C}_1+{}^5{C}_1)+{}^4{C}_2({}^3{C}_1+{}^5{C}_1)+{}^5{C}_2({}^3{C}_1+{}^4{C}_1)$

$\Rightarrow$ No of ways $=60+3(4+5)+6(3+5)+10(3+4)$

$\Rightarrow$ No of ways $=60+27+48+70=205.$

Hence, the answer is $205.$