NIMCET 2009 — Mathematics PYQ

If no three points were collinear, the total number of triangles formed by $10$ points would be:

2. Account for collinear points:

Collinear points lie on the same straight line. Selecting any $3$ points from the $6$ collinear points will not form a triangle; it will only form a line segment. We must subtract these "failed" triangles from the total.

3. Apply the Formula:

4. Step-by-Step Calculation:

• Calculate $\left(\genfrac{}{}{0ex}{}{10}{3}\right)$:

  $$\left(\genfrac{}{}{0ex}{}{10}{3}\right)=\frac{10\times 9\times 8}{3\times 2\times 1}=10\times 3\times 4=120$$

• Calculate $\left(\genfrac{}{}{0ex}{}{6}{3}\right)$:

  $$\left(\genfrac{}{}{0ex}{}{6}{3}\right)=\frac{6\times 5\times 4}{3\times 2\times 1}=5\times 4=20$$

• Subtract the values:

  $$\text{Number of Triangles}=120-20=100$$

Final Answer:

The number of triangles formed is 100. The correct option is (a).