Question

The sides $AB,BC,CA$ of a triangle$ABC$ have $3,4$ and $5$ interior points respectively on them the total no. of triangles that can be constructed by using these points as vertices are?

• A. $220$
• B. $204$
• C. $205$
• D. $195$

Answer 1

The correct option is C

$205$

Explanation for the correct option.

Find the number of triangles.

Total number of points $=3+4+5=12$.

Three collinear points do not form triangles together, so we have to remove the combination of three collinear points from the final answer.

Total number of triangles formed

$={}^{12}{C}_3-\left({}^{3}C_3+{}^{4}C_3+{}^{5}C_3\right)$

$$\begin{gathered} =220-\left(1+4+10\right) \\ =205 \end{gathered}$$

Hence, option $C$ is the correct answer.