2008 Gauss 8 Problem 22

A 3 \times 3 grid is filled with the digits 1, 2 and 3 so that each number appears once in in each row and column. Two different examples are \begin{array}{|c|c|c|}\hline 1 & 2 & 3 \\ \hline 3 & 1 & 2 \\ \hline 2 & 3 & 1 \\ \hline \end{array} \quad \text{and} \quad \begin{array}{|c|c|c|}\hline 3 & 2 & 1 \\ \hline 2 & 1 & 3 \\ \hline 1 & 3 & 2 \\ \hline \end{array}. How many different ways are there of filling the grid?

\textbf{(A)}\ 14\quad \textbf{(B)}\ 12\quad \textbf{(C)}\ 10\quad \textbf{(D)}\ 8\quad \textbf{(E)}\ 6

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.