A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelogram then formed is

Options

A. `("^m C_2)^2`
B. `("^(m+1)C_2)^2`
C. `("^(m+2)C_2)^2`
D. none of these

Answer

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

### Step-by-Step Solution

1. **Understanding the Setup**:
- We have a parallelogram that is cut by \( m \) lines parallel to one pair of its sides (let's say the horizontal sides) and \( m \) lines parallel to the other pair of sides (the vertical sides).
- This means we will have \( m + 2 \) horizontal lines (the original two sides of the parallelogram plus \( m \) additional lines) and \( m + 2 \) vertical lines.

2. **Choosing Lines to Form Parallelograms**:
- To form a parallelogram, we need to select 2 horizontal lines and 2 vertical lines.
- The number of ways to choose 2 lines from \( m + 2 \) horizontal lines is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of lines and \( r \) is the number of lines to choose. Thus, the number of ways to choose 2 horizontal lines is:
\[ \binom{m + 2}{2} \]
- Similarly, the number of ways to choose 2 vertical lines from \( m + 2 \) vertical lines is also:
\[ \binom{m + 2}{2} \]

3. **Calculating the Total Number of Parallelograms**:
- Since the selection of horizontal lines and vertical lines are independent, we multiply the number of ways to choose the horizontal lines by the number of ways to choose the vertical lines:
\[ \text{Total Parallelograms} = \binom{m + 2}{2} \times \binom{m + 2}{2} \]
- This can be simplified as:
\[ \text{Total Parallelograms} = \left( \binom{m + 2}{2} \right)^2 \]

4. **Final Expression**:
- The final expression for the total number of parallelograms formed is:
\[ \left( \frac{(m + 2)(m + 1)}{2} \right)^2 \]

### Conclusion
Thus, the total number of parallelograms formed is given by:
\[ \left( \binom{m + 2}{2} \right)^2 \]