The parallelogram is cut by two sets of m lines parallel to its sides. The numbers parallelogram thus 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 given 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 two sets of \( m \) lines.
- One set of lines is parallel to one pair of sides, and the other set is parallel to the other pair of sides.

2. **Counting the Lines**:
- For the lines parallel to one pair of sides, there are \( m \) lines plus the two original sides of the parallelogram. Therefore, the total number of lines in this direction is:
\[ m + 2 \]
- Similarly, for the lines parallel to the other pair of sides, we also have:
\[ m + 2 \]

3. **Choosing Lines to Form Parallelograms**:
- A parallelogram is formed by choosing 2 lines from the set of lines parallel to one pair of sides and 2 lines from the set of lines parallel to the other pair of sides.
- The number of ways to choose 2 lines from \( m + 2 \) lines is given by the combination formula \( C(n, r) \), which is:
\[ C(m + 2, 2) \]

4. **Calculating Total Parallelograms**:
- Since we need to choose 2 lines from both sets, the total number of parallelograms formed is:
\[ C(m + 2, 2) \times C(m + 2, 2) \]
- This can be expressed as:
\[ \left( C(m + 2, 2) \right)^2 \]

5. **Final Expression**:
- Therefore, the total number of parallelograms formed is:
\[ \left( C(m + 2, 2) \right)^2 \]

### Conclusion:
The correct answer is \( C(m + 2, 2)^2 \).