Starting with the letter A, it can be written in any of the nine spaces on the board. Once the letter $\mathbf{A}$ is written, there are six spaces left where the letter B can be written. Once the letters $\mathbf{A}$ and $\mathbf{B}$ are written on the board, there are three spaces left for the letter $\mathbf{C}$ to be written. Thus, by the multiplicative principle, there are $9 \times 6 \times 3=162$ different ways for the letters $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ to be written on the board.
A square board of three rows by three columns contains nine cells. In how many different ways can we write the three letters A, B, and $\mathbf{C}$ in three different cells, so that in each row exactly one of these three letters is written?