Count the number of rectangles of any size embedded in the given figure and…

Solution: Count all rectangles in the grid.

Step 1: Identify the grid size. The figure is a grid with 3 columns and 3 rows, so m = 3 and n = 3.

Step 2: Use the formula for the number of rectangles in an m-by-n grid:

Number of rectangles = m(m+1)/2 × n(n+1)/2.

Step 3: Compute the values.

• For columns: m(m+1)/2 = 3×4/2 = 6.

• For rows: n(n+1)/2 = 3×4/2 = 6.

• Total rectangles = 6 × 6 = 36.

Optional breakdown by rectangle size (width × height):

• 1×1: 3×3 = 9

• 2×1 (or 1×2): 2×3 = 6 and 3×2 = 6

• 2×2: 2×2 = 4

• 3×1 (or 1×3): 1×3 = 3 and 3×1 = 3

• 2×3: 2×1 = 2; 3×2: 1×2 = 2; 3×3: 1×1 = 1

If you add these counts (9 + 6 + 3 + 6 + 4 + 2 + 3 + 2 + 1) you get 36, confirming the result.

Answer: 36