The number of triangles that can be formed by choosing the vertices from a set…

Concept: A triangle needs three points that are not all on one straight line. So, among a set of points, the number of triangles = (ways to choose any 3 points) minus (ways to choose 3 points that all lie on the same line, since those do not form a triangle). If there are n points in total and m of them are collinear (with no other 3 points collinear), the number of triangles = nC3 minus mC3.

Application: Here n = 12 (total points) and m = 7 (points lying on the line).

1. Total ways to choose any 3 of the 12 points: 12C3 = (12 x 11 x 10) / (3 x 2 x 1) = 220.

2. Ways to choose 3 points that all lie on the line (these 3 points are collinear and cannot form a triangle): 7C3 = (7 x 6 x 5) / (3 x 2 x 1) = 35.

3. Number of triangles = 220 - 35 = 185.

Cross-check: Split the 12 points into the 5 points that are off the line and the 7 points that are on the line, and count triangles case by case by how many of the 3 chosen vertices come from each group (a valid triangle needs at least one vertex off the line):

• All 3 vertices off the line: 5C3 = 10.

• 2 vertices off the line, 1 vertex on the line: 5C2 x 7C1 = 10 x 7 = 70.

• 1 vertex off the line, 2 vertices on the line: 5C1 x 7C2 = 5 x 21 = 105.

Adding the three cases: 10 + 70 + 105 = 185, which matches the result from the direct subtraction method above, confirming 185 as the count.