Seven parallel horizontal lines intersect six parallel vertical lines in a plane. How many different parallelograms can be formed using these lines as the sides of the parallelograms?

Introduction / Context:
This question involves counting parallelograms formed by a grid of parallel lines. Parallelograms are determined by selecting two distinct horizontal lines and two distinct vertical lines. Each pair of horizontal lines and each pair of vertical lines forms a unique parallelogram. Recognizing this pattern and using combinations reduces the geometry question to a simple counting problem.

Given Data / Assumptions:
Number of horizontal parallel lines = 7. Number of vertical parallel lines = 6. All horizontal lines are parallel to each other and distinct. All vertical lines are parallel to each other and distinct. Horizontal lines are perpendicular to vertical lines. Any two distinct horizontal lines and two distinct vertical lines determine a parallelogram.

Concept / Approach:
A parallelogram on this grid is defined by choosing two distinct horizontal lines for the top and bottom, and two distinct vertical lines for the left and right sides. Since lines within each family (horizontal or vertical) are parallel, any two horizontal lines combined with any two vertical lines create exactly one unique parallelogram. Therefore, the total number of parallelograms is the product of the number of ways to choose horizontal pairs and the number of ways to choose vertical pairs.

Step-by-Step Solution:
Step 1: Compute the number of ways to choose 2 horizontal lines from 7. Step 2: Number of horizontal pairs = 7C2 = 7 * 6 / 2 = 21. Step 3: Compute the number of ways to choose 2 vertical lines from 6. Step 4: Number of vertical pairs = 6C2 = 6 * 5 / 2 = 15. Step 5: For each horizontal pair, any vertical pair forms a parallelogram. Step 6: Total parallelograms = 21 * 15. Step 7: Calculate 21 * 15 = 315. Step 8: Therefore, 315 distinct parallelograms can be formed.

Verification / Alternative check:
A rough check is to see that the number of horizontal pairs 21 and vertical pairs 15 are both positive and moderate. Their product is 315, which is comfortably below the total number of rectangles that could be formed if more directions were allowed. Also, if we had one fewer vertical line (5), the number of parallelograms would be 7C2 * 5C2 = 21 * 10 = 210, and adding another vertical line reasonably increases the count to 315. The arithmetic 21 * 15 = 315 is straightforward and confirms the result.

Why Other Options Are Wrong:
The value 215 may come from a miscalculation of 7C2 or 6C2 or an incorrect addition instead of multiplication. The value 415 is larger than the correct product and has no natural combinatorial justification here. The value 115 is far too small for a grid with 7 by 6 lines. Only 315 corresponds exactly to 7C2 * 6C2 and matches the logical structure of choosing two horizontal and two vertical lines.

Common Pitfalls:
Students sometimes confuse the number of rectangles or squares with the number of parallelograms in such problems. In this configuration, every rectangle is also a parallelogram, but the counting method remains the same: pairs of lines in each direction. Another mistake is to try to count parallelograms by visual inspection, which quickly becomes messy. Others may forget that the lines are infinite and not counted by segments. Using combinations for line pairs is the clean and reliable method.

Final Answer:
The total number of parallelograms formed by the 7 horizontal and 6 vertical lines is 315.