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 determine the number of different straight lines that can be formed by joining any two of the 12 points, where 7 of these points lie on the same straight line, we will follow a step-by-step approach:

1. **Calculate the total number of lines formed by joining any two points out of 12 points**:
   This can be done using the combination formula \( \binom{n}{2} \), where \( n \) is the number of points. For 12 points:
   \[
   \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66
   \]

2. **Subtract the lines formed only by the 7 collinear points**:
   For the 7 collinear points, the number of lines formed by joining any two of these points is:
   \[
   \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21
   \]
   However, all 7 collinear points lie on the same line, so these 21 combinations correspond to only one line.

3. **Account for lines formed by each pair of collinear points (already considered)**:
   We have already counted these lines as part of the total lines. We just need to recognize they represent only one unique line.

4. **Total unique lines**:
   Now, subtract the combinations of the 7 collinear points and add the unique line from these points:
  

Explore how to calculate the number of different straight lines formed by joining any two of twelve points on a plane, with seven points lying on the same line. This problem involves combinatorics and geometry concepts, using the combination formula to determine total and collinear lines. Suitable for advanced high school students.

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