Find the number of straight lines that can be drawn through any two points 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 drawn through any two points out of 10 points, of which 7 are collinear, we can follow these steps:

### Step 1: Calculate the total number of ways to choose 2 points from 10 points.
The total number of ways to choose 2 points from 10 points can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 10 \) and \( r = 2 \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

### Step 2: Calculate the number of ways to choose 2 points from the 7 collinear points.
Since the 7 points are collinear, any two points chosen from these will form the same line. Thus, the number of ways to choose 2 points from 7 points is:
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 3: Adjust the total number of lines.
From the total number of lines calculated in Step 1, we need to subtract the number of lines formed by the collinear points (since they all lie on the same line) and add 1 for the single line formed by the 7 collinear points:
\[ \text{Total lines} = \binom{10}{2} - \binom{7}{2} + 1 \]
Substituting the values we calculated:
\[ \text{Total lines} = 45 - 21 + 1 = 25 \]

### Conclusion:
Thus, the total number of straight lines that can be drawn through any two points out of the 10 points, of which 7 are collinear, is **25**.

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