Discussion
In a 2 × 4 rectangle grid shown below, each cell is a rectangle. How many rectangles can be observed in the grid?
Answer: C
30 rectangles can be observed.
A 2 × 4 grid is drawn with 3 horizontal lines and 5 vertical lines. Every rectangle in it is fixed by choosing two of the horizontal lines and two of the vertical lines:
\[ \binom{3}{2} \times \binom{5}{2} = 3 \times 10 = 30 \]
For example, the first row alone contains \(\binom{5}{2} = 10\) rectangles, the second row 10 more, and rectangles spanning both rows another 10. 21 and 27 miss some of these, and 36 is the count for a grid of 3 × 3 cells, not 2 × 4.
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