There are n-points `(ngt2)` in each to two parallel lines. Every point on one line is joined to every point on the other line by a line segment drawn within the lines. The number of point (between the lines) in which these segments intersect, is

Options

A. `""^(2n)C_(2)-2.""^(n)C_(1)+2`
B. `""^(2n)C_(2)-2xx""^(n)C_(2)`
C. `""^(n)C_(2)xx""^(n)C_(2)`
D. none of these

Answer

To solve the problem, we need to find the number of intersection points formed by line segments connecting points on two parallel lines. Let's break down the solution step by step.

### Step 1: Understanding the Setup
We have two parallel lines, each containing \( n \) points. We will denote the points on the first line as \( A_1, A_2, \ldots, A_n \) and the points on the second line as \( B_1, B_2, \ldots, B_n \).

### Step 2: Connecting the Points
Each point on the first line is connected to every point on the second line by a line segment. This means that for every point \( A_i \) on the first line, there are \( n \) segments connecting it to each of the \( B_j \) points on the second line.

### Step 3: Identifying Intersection Points
An intersection occurs between two segments \( A_iB_j \) and \( A_kB_l \) if and only if the points are chosen such that:
- \( i < k \) and \( j > l \) or
- \( i > k \) and \( j < l \)

This means that to create an intersection, we need to choose two points from each line.

### Step 4: Choosing Points
To find the number of ways to select two points from each line, we can use combinations. The number of ways to choose 2 points from \( n \) points is given by \( \binom{n}{2} \).

### Step 5: Total Intersections
Since we can choose any two points from the first line and any two points from the second line, the total number of intersection points is given by:
\[ \text{Total Intersections} = \binom{n}{2} \times \binom{n}{2} \]
This simplifies to:
\[ \text{Total Intersections} = \left( \binom{n}{2} \right)^2 \]

### Step 6: Final Expression
The final expression for the number of intersection points between the segments is:
\[ \left( \frac{n(n-1)}{2} \right)^2 = \frac{n^2(n-1)^2}{4} \]

### Conclusion
Thus, the number of points between the lines where the segments intersect is \( \binom{n}{2}^2 \).