Number of triangles that can be formed from 12 points on a plane when 8 of them are collinear

Introduction / Context:
This geometry based combinatorics question asks you to count the number of triangles that can be formed from a set of points, some of which are collinear. Triangles require three non collinear points, so any triple of collinear points cannot form a triangle.

Given Data / Assumptions:

• There are 12 points in total on a plane.
• Out of these, 8 points are collinear, that is, they lie on the same straight line.
• The remaining 4 points are not on that line and are in general position so that no additional sets of three points are collinear.
• A triangle is formed by choosing 3 points that are not collinear.

Concept / Approach:
First, count all possible triples of points that can be chosen from the 12 points. Then, subtract the triples that do not form triangles, which are precisely the triples taken entirely from the 8 collinear points. This is a standard use of combinations with exclusion of degenerate cases.

Step-by-Step Solution:
Step 1: Count all possible triples of points from 12 points using 12C3.Step 2: Compute 12C3 = 12 * 11 * 10 / (3 * 2 * 1) = 220.Step 3: Now consider the 8 collinear points. Any three chosen from these form a straight line, not a triangle.Step 4: The number of such collinear triples is 8C3.Step 5: Compute 8C3 = 8 * 7 * 6 / (3 * 2 * 1) = 56.Step 6: Valid triangles are all triples minus collinear triples.Step 7: Number of triangles = 220 - 56 = 164.

Verification / Alternative check:
You could also think in terms of constructing triangles that include at least one of the 4 non collinear points. It is more complex but should lead to the same total. The subtraction method is much more straightforward and widely used in such problems.

Why Other Options Are Wrong:

• 220 is the total number of triples and ignores the fact that some triples are collinear and do not form triangles.
• 20 and 160 are far from the correct value and do not match any of the combination steps.

Common Pitfalls:
A common mistake is to forget that three collinear points do not form a triangle. Another error is to subtract the wrong number of collinear triples, for example using 8C2 instead of 8C3. Always ensure that you use combination counts that match the size of the sets you are excluding.

Final Answer:
The number of triangles that can be formed is 164, so the correct answer is 164.