To determine how many rectangles can be formed on a $6 \times 4$ grid where the sides of the rectangles are parallel to the sides of the grid, we need to follow these steps:

Step 1: Understand the grid

• A $6 \times 4$ grid has 7 vertical lines and 5 horizontal lines, because a $n \times m$ grid has $n+1$ vertical and $m+1$ horizontal lines.
• To form a rectangle, we need to choose two distinct vertical lines and two distinct horizontal lines.

Step 2: Calculate the number of ways to choose lines

• Choosing two vertical lines: There are 7 vertical lines, and we need to choose 2. The number of ways to choose 2 vertical lines from 7 is given by the combination: $\binom{7}{2} = \frac{7 \times 6}{2} = 21$
• Choosing two horizontal lines: Similarly, there are 5 horizontal lines, and we need to choose 2. The number of ways to choose 2 horizontal lines from 5 is: $\binom{5}{2} = \frac{5 \times 4}{2} = 10$

Step 3: Multiply the results

• The total number of rectangles is the product of these two combinations: $\text{Total rectangles} = \binom{7}{2} \times \binom{5}{2} = 21 \times 10 = 210$

Thus, the total number of rectangles that can be formed on a $6 \times 4$ grid is 210.

Would you like more details on any part of this solution?

Here are some related questions to explore:

1. How many squares can be formed on a $6 \times 4$ grid?
2. What if the grid was $8 \times 5$? How many rectangles would that grid have?
3. How many ways can you choose two lines from a set of 10 lines?
4. How do we derive the formula for combinations?
5. What are the properties of a grid that affect the number of possible rectangles?

Tip: To form a square, both the vertical and horizontal distances must be equal, so you need to focus on grids where this is possible!