There are 12 points in a plane out of which 7 are in a straight line. The number of streng formed by joining all these points is

To find the number of triangles formed by joining all 12 points in the plane, where 7 points are collinear, we can use the following steps:

### Total Number of Triangles from 12 Points

First, calculate the total number of triangles that can be formed from 12 points. This can be done using the combination formula \( \binom{n}{k} \), where \( n \) is the total number of points and \( k \) is 3 (since a triangle is formed by 3 points):

\[
\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]

### Subtracting the Collinear Points

Next, calculate the number of triangles that can be formed by the 7 collinear points. Since these points are collinear, they cannot form a triangle. We need to subtract these combinations from the total:

\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Final Number of Triangles

Subtract the number of invalid (collinear) triangles from the total number of triangles:

\[
220 - 35 = 185
\]

Thus, the number of triangles that can be formed by joining all the 12 points, where 7 points are collinear, is \( \boxed{185} \).

Would you like more details or have any questions?

Here are 8 relative questions to expand your understanding:

1. How is the combination formula \(\binom{n}{k}\) derived?
2. Why can't collinear points form a triangle?
3. What is the significance of the factorial in the combination formula?
4. How would the number of triangles change if more or fewer points were collinear?
5. Can you derive the combination formula using Pascal's Triangle?
6. How would this problem change if it involved 3D space instead of a plane?
7. What other geometric shapes can be formed by connecting these points?
8. How can we generalize this approach to \( n \) points with \( m \) collinear?

**Tip:** When dealing with combinatorial problems, always consider breaking down the problem into simpler parts and look for symmetrical properties to simplify calculations.

Discover how to calculate the number of triangles that can be formed by joining 12 points in a plane, where 7 points are collinear. Step-by-step solution using combinatorial mathematics and geometry. Suitable for high school students learning combinatorics and geometric properties.

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