Question

Out of $10$points in a plane $6$ are in a straight line. The number of triangles formed by joining these points are?

• A. $100$
• B. $150$
• C. $120$
• D. None of these

Answer 1

The correct option is A

$100$

Explanation for the correct option:

Find the number of triangles:

Number of triangles formed if all $10$ points are non-collinear $={}^{10}\mathrm{C}_3$

Given $6$ points are collinear

Number of triangles formed using $6$ collinear points $={}^6\mathrm{C}_3$

So, the required number of triangles$=$ Total number of triangles $-$number of triangles formed using $6$ collinear points

$$\begin{gathered} ={}^{10}\mathrm{C}_3-{}^6\mathrm{C}_3 \\ =\frac{10!}{7!3!}-\frac{6!}{3!3!} \\ =\frac{10\times 8\times 9}{3\times 2}-\frac{6\times 5\times 4}{3\times 2} \\ =120-20 \\ =100 \end{gathered}$$

Therefore the number of triangles formed by joining these points are $100$

Hence, option (A) is the correct answer.