In a 2 × 4 rectangle grid shown below, each cell is a rectangle. How many…

Key idea: the number of rectangles in an m×n grid equals C(m+1,2) × C(n+1,2).

Here m = 2 rows and n = 4 columns.

Compute the combinations:

• Choose vertical boundaries: C(4+1,2) = C(5,2) = 10.

• Choose horizontal boundaries: C(2+1,2) = C(3,2) = 3.

Total rectangles = 3 × 10 = 30.

Verification by counting rectangles of each size:

• 1×1 rectangles: 2 × 4 = 8.

• 1×2 rectangles: 2 × 3 = 6.

• 1×3 rectangles: 2 × 2 = 4.

• 1×4 rectangles: 2 × 1 = 2.

• 2×1 rectangles: 1 × 4 = 4.

• 2×2 rectangles: 1 × 3 = 3.

• 2×3 rectangles: 1 × 2 = 2.

• 2×4 rectangles: 1 × 1 = 1.

Sum: 8 + 6 + 4 + 2 + 4 + 3 + 2 + 1 = 30, confirming the result.