Number of different striaght lines that cn be formed by joining 12 different points on a plane of which 4 are collinear is

Options

A. 16
B. 61
C. 65
D. 37

Answer

To solve the problem of finding the number of different straight lines that can be formed by joining 12 different points on a plane, of which 4 are collinear, we will follow these steps:

### Step 1: Calculate the total number of ways to select 2 points from 12 points.
To form a straight line, we need at least 2 points. The number of ways to choose 2 points from 12 is given by the combination formula:

\[ \text{Total ways} = \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

### Step 2: Calculate the number of ways to select 2 points from the 4 collinear points.
Since 4 of the points are collinear, any line formed by these points will not be unique. The number of ways to choose 2 points from these 4 collinear points is:

\[ \text{Collinear ways} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 3: Adjust for the collinear points.
Since all 4 collinear points lie on the same line, they only contribute one unique line. Therefore, we need to subtract the 6 ways of selecting 2 points from these 4 collinear points and add back 1 for the single line they form:

\[ \text{Unique lines} = \text{Total ways} - \text{Collinear ways} + 1 = 66 - 6 + 1 = 61 \]

### Final Answer
Thus, the total number of different straight lines that can be formed is:

\[ \boxed{61} \]

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