If m parallel lines in a plane are intersected by a family of n parallel lines, then the number of parallelograms that can be formed is

Options

A. a. `m^n`
B. b. `(m+1) (n+1)`
C. c. `((m-n))/(n!)`
D. d. `(mn(m-1)(n-1))/(4)`

Answer

To solve the problem of finding the number of parallelograms that can be formed by m parallel lines intersected by n parallel lines, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have m parallel lines in one direction (let's say horizontal) and n parallel lines in another direction (let's say vertical).
- A parallelogram is formed by selecting two lines from the first set of parallel lines and two lines from the second set of parallel lines.

2. **Choosing Lines**:
- To form a parallelogram, we need to select 2 lines from the m parallel lines. The number of ways to choose 2 lines from m lines is given by the combination formula \( C(m, 2) \).
- Similarly, we need to select 2 lines from the n parallel lines, which can be done in \( C(n, 2) \) ways.

3. **Combination Formula**:
- The combination formula \( C(k, r) \) is defined as:
\[ C(k, r) = \frac{k!}{r!(k-r)!} \]
- For our case:
\[ C(m, 2) = \frac{m!}{2!(m-2)!} = \frac{m(m-1)}{2} \]
\[ C(n, 2) = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]

4. **Calculating the Total Number of Parallelograms**:
- The total number of parallelograms that can be formed is the product of the combinations:
\[ \text{Total Parallelograms} = C(m, 2) \times C(n, 2) \]
- Substituting the values we calculated:
\[ \text{Total Parallelograms} = \left( \frac{m(m-1)}{2} \right) \times \left( \frac{n(n-1)}{2} \right) \]
- Simplifying this gives:
\[ \text{Total Parallelograms} = \frac{m(m-1) \cdot n(n-1)}{4} \]

5. **Final Answer**:
- Therefore, the number of parallelograms that can be formed is:
\[ \frac{m(m-1) \cdot n(n-1)}{4} \]