Problem:

How many rectangles are there in the given figure? Discuss how you found them.

The figure provided is a grid of 6 rows and 6 columns. We will solve this step by step.

Complete Solution:

Step 1: Formula for counting rectangles in a grid

In an $m \times n$ grid, the number of rectangles that can be formed is calculated as: $\text{Total rectangles} = \binom{m+1}{2} \cdot \binom{n+1}{2}$ This formula works because:

1. Rectangles are formed by choosing two horizontal lines from the $m+1$ horizontal lines (since there are $m$ rows, there are $m+1$ horizontal grid lines).
2. Similarly, rectangles are formed by choosing two vertical lines from the $n+1$ vertical lines.

Here, the grid has $6$ rows and $6$ columns, so there are $6+1=7$ horizontal lines and $6+1=7$ vertical lines.

Step 2: Compute the number of ways to choose two lines

1. For horizontal lines, the number of ways to choose 2 lines out of 7 is: $\binom{7}{2} = \frac{7 \cdot 6}{2} = 21$
2. For vertical lines, the number of ways to choose 2 lines out of 7 is: $\binom{7}{2} = \frac{7 \cdot 6}{2} = 21$

Step 3: Total number of rectangles

The total number of rectangles is the product of the two combinations: $\text{Total rectangles} = 21 \cdot 21 = 441$

Final Answer:

The total number of rectangles in the grid is 441.

Explanation Recap:

• We used the combination formula to determine how many ways we can choose pairs of horizontal and vertical lines.
• The product of these combinations gives the total number of rectangles.

Would you like further clarification or assistance with another problem?

Related Questions:

1. How would the formula change if the grid were non-square (e.g., 5 rows and 7 columns)?
2. Can you derive the formula for counting rectangles step by step from basic principles?
3. What is the difference between counting rectangles and counting squares in a grid?
4. How many squares are present in the same 6×6 grid? (Hint: Squares are a subset of rectangles.)
5. If one line is missing from the grid, how would it affect the rectangle count?

Tip:

To count squares in a grid, consider the size of the squares (1×1, 2×2, etc.), and sum their occurrences separately.