GATE CS 2016 Set 2 QGA-09 — General Aptitude

To find the total number of rectangles in a grid of size $m \times n$ (where $m$ is the number of rows and $n$ is the number of columns), we can use the combinations of horizontal and vertical grid lines.

A $2 \times 4$ grid consists of:

• $3$ horizontal lines (which define the top and bottom boundaries of the rows)
• $5$ vertical lines (which define the left and right boundaries of the columns)

Any rectangle in this grid is uniquely formed by choosing $2$ horizontal lines and $2$ vertical lines.

The number of ways to choose $2$ horizontal lines out of $3$ is: $$\binom{3}{2} = 3$$

The number of ways to choose $2$ vertical lines out of $5$ is: $$\binom{5}{2} = \frac{5 \times 4}{2} = 10$$

Therefore, the total number of rectangles that can be observed in the grid is: $$\text{Total Rectangles} = \binom{3}{2} \times \binom{5}{2} = 3 \times 10 = 30$$

Thus, the correct option is C.