Question

There are $10$ points in a plane, out of these $6$ are collinear. If $\mathrm{N}$ is the number of triangles formed by joining these points, then

• A. $\mathrm{N}>190$
• B. $\mathrm{N}\le 100$
• C. $100<\mathrm{N}\le 140$
• D. $140\le \mathrm{N}\le 190$

Answer 1

The correct option is B

$\mathrm{N}\le 100$

Compute the required value:

Out of $10$ points $6$ are collinear.

Therefore, the number of triangles formed by joining these points are

$\begin{array}{rcl}\mathrm{N}& =& {}^{10}\mathrm{C}_3-{}^6\mathrm{C}_3\\ & \Rightarrow & \mathrm{N}=\frac{10!}{3!\times 7!}-\frac{6!}{3!\times 3!}\left[{}^{n}C_r=\frac{n!}{r!\times (n-r)!}\right]\\ & \Rightarrow & \mathrm{N}=\frac{10\times 9\times 8\times 7!}{3\times 2\times 7!}-\frac{6\times 5\times 4\times 3!}{3\times 2\times 3!}\\ & \Rightarrow & \mathrm{N}=120-20\\ & \Rightarrow & \mathrm{N}=100\end{array}$

Therefore, $\mathrm{N}\le 100$.

Hence, option (B) is the correct option.