Ah, thanks MotherJane for clarifying the language. For whatever reason, I wasn't thinking of the stations as being along the same track from A to B, which made the question seem nonsensical.
So, the question is really asking, if you have 12 things in a row, in how many ways can you choose four of them so that no two are consecutive?
The answer is 9C4. The above argument about 'restrictions' doesn't immediately make intuitive sense to me, but it does give the right answer, so I'm probably not looking at it from the best point of view. You do need to account for having three restricitions, not two, if you are selecting four things. So your formula would become (n-3)C4 = 9C4.
I looked at the problem in two different ways. If you choose any four stations from the first nine, which you can do in 9C4 ways, then insert an extra space between each station, you are certain to get a selection of four from twelve choices where no two are consecutive. And you'll never count the same set twice in this way, which makes the answer 9C4.
I also did this by breaking the problem down into cases. You know there's at least one space between each station (which I'll label S). If we start from a 'word' like:
S_S_S_S
we just need to insert five spaces to get a valid set of four stations from twelve, where no two are consecutive. We can then break down, case by case, how those spaces could be inserted. For example, if we insert five consecutive spaces somewhere, we have five places where we can do it. If we insert four spaces somewhere, and one space somewhere else, we have 5*4 ways to do it. And so on.... the answer again turns out to be 126, or 9C4.
There's probably a generating function approach for this question, and there certainly is for the dice problem above- but that's way, way beyond what's needed for GMAT math. I'll reiterate what I said before: this is not a GMAT question!
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
ianstewartgmat.com
