How many ways are there to place three rooks on an $8 \times 8$ chessboard such that the rooks are in different columns and different rows?
There are $\left(\begin{array}{l}8 \\ 3\end{array}\right)$ ways to pick which three of the eight columns we wish to pick the rooks in. Once the set of three columns is fixed, then we have eight choices for where to pick the rook in the leftmost column, then seven choices for where to pick the rook in the middle column, and finally six choices for where to pick the rook in the rightmost column. Thus, the answer is

$$
\left(\begin{array}{l}
8 \\
3
\end{array}\right) \cdot 8 \cdot 7 \cdot 6=18816 .
$$