How many triangles can be made by joining 12 points in a plane, given that seven are in one line?

To solve the problem of determining how many triangles can be formed by joining 12 points in a plane, where 7 of these points are collinear, we need to break it down into two steps: 

1. **Step 1: Total number of triangles from 12 points**  
   Normally, if no three points were collinear, the number of triangles that could be formed by any set of three points would be given by the combination formula \( \binom{n}{3} \), where \( n \) is the total number of points. For 12 points:

   \[
   \text{Total triangles} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
   \]

2. **Step 2: Subtract the invalid triangles (those from the collinear points)**  
   Since any three collinear points cannot form a triangle, we need to subtract the number of triangles that would be formed by choosing 3 points from the 7 collinear points. This is given by \( \binom{7}{3} \):

   \[
   \text{Invalid triangles} = \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
   \]

3. **Step 3: Final result**  
   The valid number of triangles is the total number of triangles minus the invalid triangles:

   \[
   \text{Valid triangles} = 220 - 35 = 185
   \]

Thus, the number of triangles that can be formed is **185**.

Would you like more details or have any questions?

Here are some related questions:
1. What is the general formula for calculating the number of triangles given \( n \) points, some of which are collinear?
2. How do we extend this problem if more than two sets of collinear points are present?
3. Can you explain the difference between combinations and permutations, and how they apply here?
4. What would the result be if all 12 points were collinear?
5. How do we handle cases where no points are collinear?

**Tip**: In combinatorics, always check for special cases like collinearity to ensure you aren't counting invalid configurations!

How many triangles can be made by joining 12 points in a plane, given that seven are in one line?

Discover how to calculate the number of triangles that can be formed by joining 12 points, where 7 are collinear. Using combinatorics and geometry, we walk through the step-by-step solution to solve this problem. Suitable for students in Grades 9-12.

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