A train on a 12-station route has to be stopped at 4 stations with no two stops consecutive. How many ways can the stops be arranged?
Let $S_1,S_2,S_3,S_4$ be the stops. $$A\to x_1\to S_1\to x_2\to S_2\to x_3\to S_3\to x_4\to S_4\to x_5\to B$$ $x_1$ is the number of stations between $A$ and $S_{1}$, and similarly for the other $x_i$'s.
I want to go further, could some help me with this?
This is a standard multiset (stars-and-bars) problem. Consider the eight stations where the train doesn't stop, which define nine gaps:
_S_S_S_S_S_S_S_S_
We want to add four more stations where the train does stop into these gaps, with at most one stop per gap to satisfy the condition of no consecutive stops. There are $\binom94=126$ ways of doing this, and this is therefore the answer.
