JEE Mathematics 2025 Question with Solution

Given: There are $12$ points in a plane, and no three are collinear except $5$ points which are collinear.

Find: The total number of triangles that can be formed.

First count all possible ways of choosing any $3$ points from $12$ points:

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

Now subtract the selections in which all $3$ points come from the $5$ collinear points, because such points do not form a triangle:

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

Therefore, the required number of triangles is:

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

Therefore, the total number of triangles is $210$. The correct option is A.

Given: Total points are $12$, and $5$ of them are collinear.

Find: The number of valid triangles.

A triangle is formed whenever the chosen $3$ points are not collinear. So count all triples and remove only the invalid collinear triples:

$$\text{Valid triangles} = \binom{12}{3} - \binom{5}{3}$$ $$= 220 - 10 = 210$$

This works because the only triples that fail to form triangles are those chosen entirely from the $5$ collinear points.

Therefore, the correct option is A.