There are 12 points in a plane of which 5 are collinear . Find the number of straight lines obtained by joining these points in pairs .

Answer

To find the number of straight lines obtained by joining pairs of 12 points in a plane, where 5 of those points are collinear, we can follow these steps:

### Step 1: Calculate the total number of lines from 12 points
The total number of ways to choose 2 points from 12 points is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose. Here, \( n = 12 \) and \( r = 2 \).

\[ \text{Total lines} = \binom{12}{2} = \frac{12 \times 11}{2} = 66 \]

### Step 2: Calculate the lines formed by the collinear points
Since 5 of the points are collinear, they will only form one line. The number of ways to choose 2 points from these 5 collinear points is:

\[ \text{Collinear lines} = \binom{5}{2} = \frac{5 \times 4}{2} = 10 \]

However, since all 5 points are on the same line, they contribute only 1 unique line instead of 10.

### Step 3: Adjust the total lines for collinear points
To find the actual number of unique lines, we need to subtract the 10 lines formed by choosing pairs from the 5 collinear points and add back 1 for the single line they actually form:

\[ \text{Unique lines} = \text{Total lines} - \text{Collinear pairs} + 1 \]

Substituting the values we calculated:

\[ \text{Unique lines} = 66 - 10 + 1 = 57 \]

### Final Answer
Thus, the total number of straight lines obtained by joining these points in pairs is **57**.

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