There are $N$ intermediate stations on a railway line from one terminus to another. In how many ways can a train stop at three of these intermediate stations if no two of these stopping stations are to be consecutive?
I observed that $N \ge 5 $ and for $N=5,6,7,8,9,10$ the answers are $1, 4, 10, 20, 35, 56$.
Then with the aid of tetrahedral numbers, I guessed that the general answer should be $ { N-2 \choose 3}$. and this happens to be correct.
I was wondering how to prove this (preferably a combinatorial way)? And what about the more general problem of the train stopping in $K$ of $N$ intermediate stations?