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.
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"Exactly one per row and per column" \Rightarrow think of permutation matrices or non-attacking rooks; count with $n!$ for an $n\times n$ grid.
Step 1: Model the constraint.
Each row must contain exactly one shaded cell and each column must contain exactly one shaded cell.
This is equivalent to placing $3$ non-attacking rooks on a $3\times3$ board—one per row and one per column.
Step 2: Interpret as a permutation.
Choose, for each row $i\in\{1,2,3\}$, the column $j$ where its single shaded cell goes.
Because columns cannot repeat, this assignment is a permutation of the three columns.
Step 3: Count the permutations.
Number of permutations of $3$ distinct columns $= 3! = 6$.
\[
\boxed{6}
\]
