Question

Choose $3,4,5$ points other than vertices respectively on the sides $AB,BC$ and $CA$ of a $△ABC$. The number of triangles that can be formed by using only these points as vertices, is

• A. $220$
• B. $217$
• C. $215$
• D. $210$
• E. $205$

Answer 1

The correct option is E $205$

Required number of triangles that can be formed by using only given points as vertices
${=}^{12}{C}_3-\left\{{}^3{C}_3{+}^4{C}_3{+}^5{C}_3\right\}$
$=\frac{12\times 11\times 10}{3\times 2\times 1}-\{1+4+\frac{5\times 4}{2\times 1}\}$
$=220-(5+10)$
$=220-15=205$