There are 12 points in a plane, out of which 3 points are collinear. How many straight lines can be drawn by joining any two of them?

Explanation

To solve the problem of how many straight lines can be drawn by joining any two of the 12 points in a plane, where 3 of those points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the total number of ways to select 2 points from 12 points:**
The total number of lines that can be formed by joining any two points from 12 points is given by the combination formula \( nCk \), where \( n \) is the total number of points and \( k \) is the number of points to choose (in this case, 2).
\[ \text{Total Lines} = \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

2. **Calculate the number of lines formed by the 3 collinear points:**
Since the 3 points are collinear, they do not form distinct lines when taken in pairs. Instead, they only form 1 line. The number of ways to choose 2 points from these 3 collinear points is:
\[ \text{Collinear Lines} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = 3 \]
However, since these 3 points are collinear, they only contribute 1 line, not 3.

3. **Adjust the total lines for collinear points:**
To find the actual number of distinct lines, we subtract the excess lines formed by the collinear points. We subtract the 3 lines that would have been counted and add back 1 for the single line that they actually form:
\[ \text{Distinct Lines} = \text{Total Lines} - \text{Collinear Lines} + 1 = 66 - 3 + 1 = 64 \]

### Final Answer:
The total number of distinct straight lines that can be drawn by joining any two of the 12 points is **64**.

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