6. Abelarda, Brunilda, and Callisto, three old acquaintances, want to buy a house each among the 10 cottages in a row on the main street of the city. Since they can't stand each other, they absolutely want to avoid being neighbors: they therefore wish that the houses they buy are not adjacent to each other. In how many ways can they buy houses to satisfy this condition?
(A) 56
(B) 120
(C) 336
(D) 480
(E) 504
6. The answer is $\mathbf{( C )}$. First, consider a slightly different problem: leave the number of houses undetermined for now, and ignore the assumption that the houses cannot be adjacent. In this case, if $n$ is the number of houses on the street, we would have $\left(\begin{array}{l}n \\ 3\end{array}\right)$ ways to choose which houses are inhabited by the three acquaintances and 3! ways to assign these houses to the three.

Now, to ensure that the houses are not adjacent, we just need to insert a house after the first choice and another after the second choice. Since we know that there are 10 houses in total on the street, this means that the $n$ houses considered earlier are actually 8 houses. Therefore, the number we are looking for is

$$
\left(\begin{array}{l}
8 \\
3
\end{array}\right) \cdot 3 !=\frac{6 \cdot 7 \cdot 8}{3 !} \cdot 3 !=336
$$