GATE Civil 2023 Set 2 | GA Question: 3

Question

In how many ways can cells in a $3 \times 3 $ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown. 

 

• A. $2$
• B. $9$
• C. $3$
• D. $6$

Answer 1

We have a $3 \times 3$ grid, and we want to fill it in such a way that each row and each column have exactly one shaded cell.

First, we filled in the red color, second we filled in the blue color, and then, at last, we filled in the green color.

We got the total number of ways $ = 6$

$\textbf{Short Method:}$

From the above diagram,

• Red colors have $3$ choices to fill the grid.
• After that blue colors have $2$ choices to fill the grid.
• After that green colors have $1$ choices to fill the grid.

So, the total number of ways $ = 3 \times 2 \times 1 = 6$

Correct Answer: D