There are 10 points in a plane, out of which 5 are collinear. Find the number of straight lines formed by joining them.

Options

A. 36
B. 45
C. 30
D. 35

Answer

To solve the problem of finding the number of straight lines formed by joining 10 points in a plane, where 5 of those points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Points**:
- We have a total of 10 points.
- Among these, 5 points are collinear, meaning they all lie on the same straight line.

2. **Calculating Lines from Total Points**:
- To find the number of straight lines that can be formed from any two points, we use the combination formula \( nC2 \), where \( n \) is the total number of points.
- Here, \( n = 10 \).
- The number of lines formed by choosing any 2 points from 10 is given by:
\[ \text{Total Lines} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

3. **Calculating Lines from Collinear Points**:
- Since 5 points are collinear, they will only form 1 line instead of 10 lines (which would be the case if they were non-collinear).
- The number of lines formed by choosing any 2 points from these 5 collinear points is:
\[ \text{Collinear Lines} = \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]
- However, since these 5 points are collinear, they only contribute 1 unique line.

4. **Adjusting for Collinearity**:
- We need to adjust our total line count by subtracting the extra lines counted from the collinear points:
\[ \text{Adjusted Lines} = \text{Total Lines} - (\text{Collinear Lines} - 1) \]
- This gives us:
\[ \text{Adjusted Lines} = 45 - (10 - 1) = 45 - 9 = 36 \]

5. **Final Count**:
- Therefore, the total number of distinct straight lines that can be formed by joining the 10 points is **36**.

### Final Answer:
The number of straight lines formed by joining the 10 points is **36**.