There are n points in a place in which p point are collinear. How many lines can be formed from these points

Options

A. `"^(n-p)C_2`
B. `"^n C_2`
C. `""^n C_2-"^P C_2+1`
D. none of these

Answer

To solve the problem of how many lines can be formed from \( n \) points in which \( p \) points are collinear, we can follow these steps:

### Step 1: Understand the Total Lines from \( n \) Points
First, we need to determine how many lines can be formed from \( n \) points without considering any collinearity. The formula to calculate the number of lines that can be formed from \( n \) points is given by the combination formula:

\[ \text{Total Lines} = \binom{n}{2} = \frac{n(n-1)}{2} \]

This formula counts all possible pairs of points, each of which can form a line.

### Step 2: Adjust for Collinear Points
Next, we need to account for the \( p \) collinear points. When points are collinear, they do not form distinct lines between each other. Instead, all \( p \) collinear points form only one line.

The number of lines that can be formed by the \( p \) collinear points is:

\[ \text{Collinear Lines} = \binom{p}{2} = \frac{p(p-1)}{2} \]

However, since these \( p \) points only form one line, we need to subtract the \( \binom{p}{2} \) lines from the total lines calculated in Step 1.

### Step 3: Calculate the Final Number of Lines
Now, we can find the total number of distinct lines formed by subtracting the lines formed by the collinear points from the total lines calculated in Step 1:

\[ \text{Distinct Lines} = \binom{n}{2} - \left(\binom{p}{2} - 1\right) \]

This simplifies to:

\[ \text{Distinct Lines} = \frac{n(n-1)}{2} - \left(\frac{p(p-1)}{2} - 1\right) \]

### Step 4: Final Formula
Thus, the final formula for the number of distinct lines that can be formed from \( n \) points in which \( p \) points are collinear is:

\[ \text{Distinct Lines} = \frac{n(n-1)}{2} - \frac{p(p-1)}{2} + 1 \]

### Summary
In conclusion, the number of lines that can be formed from \( n \) points where \( p \) points are collinear is given by:

\[ \text{Distinct Lines} = \frac{n(n-1) - p(p-1) + 2}{2} \]