JEE Main 2025 — 8 April, Evening Shift — Question 40

To find the number of valid triangles, we can calculate the total unrestricted ways to select three points and subtract the cases where the selected points form a straight line instead of a triangle.

Total ways to select 3 points out of 12: If there were no constraints, the number of ways to pick any 3 vertices from the 12 points is given by:

$$\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \text{ ways}$$

Number of invalid selections (collinear sets): Since 5 points lie on the same straight line, selecting any 3 points from these 5 will not form a triangle. The number of such straight-line combinations is:

$$\binom{5}{3} = \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \text{ ways}$$

Total Valid Triangles Subtracting the invalid cases from the total possible selections:

$$\text{Total Triangles} = \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210$$