Question

The number of rectangles in a 6 $\times$ 6 grid is ______.

• A. 441
• B. 784
• C. 1365
• D. 1008
• E. 204

Hint:

To find Rectangles, use the formula $\left( \frac{n(n+1)}{2} \right)^2$. To find Squares, use the formula $\frac{n(n+1)(2n+1)}{6}$. Rectangles always outnumber squares!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a grid of size $n \times n$, there are $(n+1)$ horizontal lines and $(n+1)$ vertical lines. We select 2 from each set to form a rectangle.

Step 2: Key Formula or Approach:
Number of rectangles in an $n \times n$ grid: \[ \left( \frac{n(n+1)}{2} \right)^2 \]

Step 3: Detailed Explanation:
1. For a $6 \times 6$ grid, $n = 6$. 2. This means there are $6 + 1 = 7$ horizontal lines and $6 + 1 = 7$ vertical lines. 3. Total rectangles = $^7C_2 \times ^7C_2$. \[ ^7C_2 = \frac{7 \times 6}{2 \times 1} = 21. \] 4. Total = $21 \times 21 = 441$.

Step 4: Final Answer:
The total number of rectangles is 441.