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.