There are n straight lines in a plane, no two of which are parallel and no three pass through the same point. Their points of intersection are joined. The number of fresh lines thus formed is

Options

A. `(n(n-1)(n-2)(n-3))/(8)`
B. `(n(n+1)(n-2)(n-3))/(8)`
C. `(n(n+1)(n+2)(n-3))/(8)`
D. `(n(n+1)(n+2)(n+3))/(8)`

Answer

To solve the problem of finding the number of fresh lines formed by the intersection points of \( n \) straight lines in a plane, where no two lines are parallel and no three lines intersect at the same point, we can follow these steps:

### Step 1: Determine the number of intersection points
When \( n \) lines are drawn in a plane, the number of intersection points formed by these lines can be calculated using the combination formula. Since no two lines are parallel and no three lines meet at a single point, every pair of lines will intersect exactly once. The number of ways to choose 2 lines from \( n \) lines is given by:

\[ \text{Number of intersection points} = \binom{n}{2} = \frac{n(n-1)}{2} \]

### Step 2: Determine the number of fresh lines formed
Each intersection point can be connected to other intersection points to form new lines. To find the number of fresh lines formed by joining these intersection points, we need to consider how many ways we can choose 2 intersection points to form a line.

The total number of intersection points is \( \binom{n}{2} \). To find the number of fresh lines formed by these points, we need to choose 2 points from the total intersection points, which can be calculated as:

\[ \text{Number of fresh lines} = \binom{\binom{n}{2}}{2} \]

### Step 3: Calculate the number of fresh lines
Using the formula for combinations, we can express this as:

\[ \text{Number of fresh lines} = \frac{\binom{n}{2} \left( \binom{n}{2} - 1 \right)}{2} \]

Substituting \( \binom{n}{2} = \frac{n(n-1)}{2} \):

\[ = \frac{\frac{n(n-1)}{2} \left( \frac{n(n-1)}{2} - 1 \right)}{2} \]

This simplifies to:

\[ = \frac{n(n-1)}{2} \cdot \frac{n(n-1) - 2}{4} \]

### Step 4: Final simplification
Continuing from the previous step, we can simplify further:

\[ = \frac{n(n-1)(n(n-1) - 2)}{8} \]

This gives us the final expression for the number of fresh lines formed:

\[ = \frac{n(n-1)(n^2 - 3n + 2)}{8} \]

### Conclusion
Thus, the number of fresh lines formed by joining the intersection points of \( n \) lines in a plane is:

\[ \frac{n(n-1)(n^2 - 3n + 2)}{8} \]