The number of straight lines can be formed out of 10 points of which 7 are collinear?

Options

A. 26
B. 21
C. 25
D. None of these

Answer

To find the number of straight lines that can be formed from 10 points, where 7 of those points are collinear, we can follow these steps:

### Step 1: Understand the Problem
We have a total of 10 points, out of which 7 points are collinear (meaning they lie on the same straight line). The remaining 3 points are not collinear with the 7 points.

### Step 2: Calculate Lines from Non-Collinear Points
First, we can form lines using any 2 points from the 10 points. The formula for selecting 2 points from n points is given by the combination formula:
\[ \text{Number of lines} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
For our case, we need to calculate \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]
This means that if all points were non-collinear, we could form 45 lines.

### Step 3: Subtract Lines Formed by Collinear Points
Next, we need to account for the fact that the 7 collinear points only form 1 line, not 21 lines (which would be the case if they were not collinear). The number of lines that can be formed using the 7 collinear points is:
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]
However, since all these points are on the same line, they only contribute 1 line.

### Step 4: Calculate the Total Number of Unique Lines
Now, we can calculate the total number of unique lines formed by subtracting the extra lines counted from the collinear points:
\[ \text{Total lines} = \text{Lines from all points} - \text{Lines from collinear points} + 1 \]
\[ \text{Total lines} = 45 - 21 + 1 = 25 \]

### Final Answer
Thus, the total number of straight lines that can be formed from the 10 points, where 7 are collinear, is **25**.

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