A parallelogram is cut by two set of n parallel lines, parallel to the sides of the parallelo-gram. The number of parallelograms formed is

Options

A. `(""^(n +2)C_(2))(""^(n+2)C_(2))`
B. `(""^(n)C_(2))(""^(n+1)C_(2))`
C. `(""^(n)C_(2))(""^(n+2)C_(2))`
D. `(""^(n)C_(2))(""^(n)C_(2))`

Answer

To solve the problem of finding the number of parallelograms formed when a parallelogram is cut by two sets of \( n \) parallel lines (each set parallel to one of the sides of the parallelogram), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a parallelogram with two sets of \( n \) parallel lines.
- One set of \( n \) lines is parallel to one pair of opposite sides of the parallelogram.
- The other set of \( n \) lines is parallel to the other pair of opposite sides.

2. **Counting Total Lines**:
- The original sides of the parallelogram contribute 2 lines (one for each side).
- Thus, the total number of lines parallel to one set of sides becomes \( n + 2 \) (the \( n \) lines plus the 2 original sides).
- Similarly, for the other set of sides, we also have \( n + 2 \) lines.

3. **Choosing Lines to Form Parallelograms**:
- To form a parallelogram, we need to select 2 lines from the first set (which has \( n + 2 \) lines) and 2 lines from the second set (which also has \( n + 2 \) lines).
- The number of ways to choose 2 lines from \( n + 2 \) lines is given by the combination formula \( \binom{n + 2}{2} \).

4. **Calculating the Total Number of Parallelograms**:
- Therefore, the total number of parallelograms formed is the product of the combinations from both sets:
\[ \text{Total Parallelograms} = \binom{n + 2}{2} \times \binom{n + 2}{2} \]

5. **Final Expression**:
- This can be simplified to:
\[ \text{Total Parallelograms} = \left( \binom{n + 2}{2} \right)^2 \]

### Conclusion:
The number of parallelograms formed is \( \binom{n + 2}{2} \times \binom{n + 2}{2} \).