Question

There are 4 horizontal and 4 vertical lines on a board. What is the maximum number of rectangles that can be formed?

• A. 16
• B. 24
• C. 36
• D. 42

Hint:

This formula works for any grid. For a $3 \times 3$ grid of squares, there are 4 horizontal and 4 vertical lines, which is exactly what this question describes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To form a rectangle, we need to select 2 horizontal lines (to form the top and bottom) and 2 vertical lines (to form the left and right sides).

Step 2: Key Formula or Approach:
If there are $m$ horizontal lines and $n$ vertical lines, the number of rectangles is: \[ ^mC_2 \times ^nC_2 \]

Step 3: Detailed Explanation:
1. Number of horizontal lines ($m$) = 4. Ways to choose 2: \[ ^4C_2 = \frac{4 \times 3}{2 \times 1} = 6. \] 2. Number of vertical lines ($n$) = 4. Ways to choose 2: \[ ^4C_2 = \frac{4 \times 3}{2 \times 1} = 6. \] 3. Total rectangles = $6 \times 6 = 36$.

Step 4: Final Answer:
The maximum number of rectangles is 36.