The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is

Options

A. 6
B. 18
C. 12
D. 9

Answer

To find the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formation of a Parallelogram**:
A parallelogram is formed by selecting two pairs of parallel lines. One pair will come from the first set of parallel lines, and the other pair will come from the second set.

2. **Identify the Sets of Lines**:
- We have 4 parallel lines in one direction (let's call this set A).
- We have 3 parallel lines in the other direction (let's call this set B).

3. **Select Lines from Set A**:
To form one pair of sides of the parallelogram, we need to select 2 lines from the 4 lines in set A. The number of ways to choose 2 lines from 4 can be calculated using the combination formula:
\[ \text{Number of ways to choose 2 from 4} = \binom{4}{2} \]

4. **Calculate \(\binom{4}{2}\)**:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

5. **Select Lines from Set B**:
Similarly, we need to select 2 lines from the 3 lines in set B. The number of ways to choose 2 lines from 3 is:
\[ \text{Number of ways to choose 2 from 3} = \binom{3}{2} \]

6. **Calculate \(\binom{3}{2}\)**:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

7. **Calculate the Total Number of Parallelograms**:
The total number of parallelograms that can be formed is the product of the number of ways to choose the lines from both sets:
\[ \text{Total parallelograms} = \binom{4}{2} \times \binom{3}{2} = 6 \times 3 = 18 \]

### Final Answer:
The number of parallelograms that can be formed is **18**.

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