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

To find the number of straight lines formed by joining 12 points in a plane where 7 points are collinear, we need to consider two main components: lines formed by collinear points and lines formed by other combinations of points.

### Step-by-Step Solution

1. **Lines formed by the 7 collinear points:**
   
   Any two of the 7 collinear points will form a line. The number of ways to choose 2 points from 7 is given by the combination formula \( \binom{7}{2} \):

   \[
   \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21
   \]

   However, these 21 combinations represent only one unique line since all 7 points are collinear.

2. **Lines formed by the remaining 5 points:**
   
   Any two of the remaining 5 points can form a line. The number of ways to choose 2 points from 5 is given by the combination formula \( \binom{5}{2} \):

   \[
   \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
   \]

3. **Lines formed by combining collinear and non-collinear points:**

   Any point from the 5 non-collinear points can be joined with any of the 7 collinear points to form a line. This is given by the product of their counts:

   \[
   7 \times 5 = 35
   \]

### Total Number of Lines

Summing up all the lines:

\[
1 \text{ (unique line from 7 collinear points)} + 10 \text{ (lines from the remaining 5 points)} + 35 \text{ (lines formed between collinear and non-collinear points)}
\]

\[
1 + 10 + 35 = 46
\]

Thus, the number of straight lines formed by joining all 12 points, where 7 points are collinear, is \( \boxed{46} \).

Would you like more details or have any questions?

Here are 8 relative questions to expand your understanding:

1. What is the general formula for finding the number of lines formed by \( n \) points in a plane?
2. How does the presence of collinear points affect the total number of lines?
3. What is the significance of collinear points in geometric constructions?
4. How would the result change if only 6 points were collinear instead of 7?
5. Can you apply this approach to find the number of lines in 3-dimensional space?
6. How does the combination formula help in solving problems related to point arrangements?
7. What are some real-world applications of understanding collinear points and line formations?
8. How would you approach finding the number of triangles formed if more than one set of collinear points existed?

**Tip:** When working with combinatorial geometry problems, always separate and categorize different sets of points based on their properties, such as collinearity, to simplify your calculations.

Explore how to calculate the number of straight lines formed by joining 12 points in a plane, where 7 points are collinear. Learn about combinatorial geometry, the impact of collinear points, and step-by-step solutions with detailed calculations. Suitable for high school students and those interested in geometric constructions.

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