Question

how many triangles can be obtained 12 points 5 collinear

Answer 1

Total number of triangles that can be formed with 12 points (if none of them are collinear) = 12C3 
(this is because we can select any three points and form the triangle if they are not collinear)

With collinear points, we cannot make any triangle (as they are in straight line).
Here 5 points are collinear. Therefore we need to subtract 5C3 triangles from the above count.

Hence, required number of triangles = 12C3 - 5C3 = 220 - 10 = 210

Answer 2

210 ways are there.

Case 1 - All three points selected from the 7 non-collinear points. This can be done in 7C3 ways. = 35 ways.

Case 2 - Two points are selected from the 7 non-collinear points, and one from the 5 collinear points.This can be done in 7C2 x 5C1  ways. = 21 x 5 = 105 ways.

Case 3 - One point is selected from the 7 non-collinear points and two points are selected from the 5 collinear points. This can be done in 7C1 x 5C2  ways. = 7 x 10 = 70 ways.

Case 4 - zero points from the 7 non-collinear points and three from the 5 collinear points. This case will not yield a traingle and only yield straight lines. So it will 0.

Hence answer is where case 1 or case 2 or case 3 happens to form a valid triangle, i.e. 35 + 105 + 70 = 210 ways.