Out of 18 points in as plane, no three points are in the same straight line except five points which are collinear. The number of straight lines formed by joining them is

Options

A. 144
B. 153
C. 152
D. 140

Explanation

To solve the problem of finding the number of straight lines formed by joining 18 points in a plane, where no three points are in the same straight line except for five collinear points, we can follow these steps:

### Step 1: Understand the total points and collinear points
We have a total of 18 points, and out of these, 5 points are collinear. This means that while we can form lines from all 18 points, the 5 collinear points will not contribute additional unique lines beyond what is formed by choosing any two of them.

### Step 2: Calculate the total lines from 18 points
The number of lines that can be formed by choosing any 2 points from 18 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.

\[ \text{Total lines from 18 points} = \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153 \]

### Step 3: Calculate the lines from the 5 collinear points
Since the 5 points are collinear, the number of lines that can be formed by choosing any 2 points from these 5 points is:

\[ \text{Lines from 5 collinear points} = \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Adjust for the collinear points
Since the 5 collinear points only form one unique line, we need to subtract the lines formed by these 5 points from the total lines calculated in Step 2. However, we must add back 1 for the single line formed by the 5 collinear points.

Thus, the total number of unique lines formed is:

\[ \text{Unique lines} = \text{Total lines from 18 points} - \text{Lines from 5 collinear points} + 1 \]

Substituting the values we calculated:

\[ \text{Unique lines} = 153 - 10 + 1 = 144 \]

### Final Answer
The total number of straight lines formed by joining the points is **144**.

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