GATE 2023 CE (CE2) – Question 3

General Aptitude · Quantitative Aptitude: Arithmetic and Number Computation · 1 mark · Multiple choice

In how many ways can cells in a $3\times3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.

3 by 3 grid with exactly one shaded cell in each row and column.

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Correct answer: (D) 6

Explanation

Choose one shaded cell in every row and every column, which is the same as placing one non-attacking rook in each row of a $3\times3$ board.

**Row 1:** any of the 3 columns.
**Row 2:** any column except the one used in row 1, so 2 choices.
**Row 3:** the one column left, so 1 choice.

Number of ways $=3\times2\times1=3!=\mathbf{6}$.