On the other hand, the probability calculation looks to me like a genuine mistake in closing, presumably because of assuming the problem was much simpler than it turns out to be. That's something we should be careful to avoid. I don't think the question is super deep or profound, and it feels like the sort of problem one ought to be able to say a lot about. However, profound depth is not necessary for a MO question, and there's a big difference between feeling like it ought to be possible to say a lot and actually being able to say it.

I would prefer if we were welcoming to researchers outside mathematics who have genuine mathematics questions, without necessarily bouncing them off M.SE first. Mathematics has a history of both renewal from other fields and fruitful interchange, but often the initial interactions are fraught with communication problems.

I agree.

There is a simple solution: MO ought to have a rule that you should not vote to close a question as too elementary unless you know what the answer is. This should be in the FAQ, IMHO.

I would actually be reluctant to put this in the FAQ: the prospect of challenges to "prove that you know how to solve this problem you voted to close" would in my view be more disruptive than occasionally having to vote to re-open a mistakenly closed question.

I certainly agree that if your reason for objecting to a question is that it's too straightforward or easy, then you ought to refrain from voting unless you are justifiably confident in your judgment, and if you make more than very rare mistakes you should reevaluate your standards. (But unless someone explicitly says straightforwardness is their reason, it's hard to deduce anything from the voting options.)

ad 4. I think this has little to do with disdain (for me, if you check what I answer you will see most of it is quite close to combinatorics in some sense, elementary NT mainly), or say, Gerry Myerson, Qiaochu Yuan and Andreas Blass, who closed the question, should disdain combinatorics? Do you really want to maintain this?

But I agree(!) that in some parts of mathematics (including CO and 'my type' of NT) there is a higher risk of errors in judgement of the form you describe (due to honest errors). [IMO the current question is a non-example for this though.]

Therefore in my opinion it is paricullarly important if one cares about these areas (as I do) to fight for a culture of context and motivation. (This is not some type of argument I construe now, if you want I can search for me saying this already really long ago on meta. I think before closing my first question even, but not sure.)

The question asked (in body and title) is that the quantity is an integer. Just this. And this is easy enough;
see Butch Malahide's answer. So it is a "simple excercise" (while the reason for thinking so was in error).

Now, that the question was motivated by a half-remembered (actually misrembered it seems) hint and some people followed up in the direction of this hint and provided interesting information is still something else and one thus might leave this open now.

But the question itself is really not a suiteable question. OP did not ask for a combinatorial interpretation/bijective proof or anything.

Please try to be more precise in your complaints. (My time and energy to point out such things are contrary to what you might think quite limited.)

Added: It might also be worth noting that Qiauchu Yuan actively questioned the comment criticised by you and still voted to close.

What a 'reasonable' and 'nice' question(er)!

[Added a couple hours later: OP seems back with a comment.]

A main problem I have with the current situation is that the user that made the effort to write a well-intentioned comment and willing to follow-up on the matter gets "all the blame" (in an IMO exaggerated way) [not by you, I find your contribution well-balanced] while those that did not, but actually closed the question, somehow are not even much addressed (not that IMO they are to be blamed for anything either but if somebody wants to blame somebody they should excercise at least a moderate amount of fairness in doing so). This is a recurring theme and in my opinion a problem.

I would prefer if we were welcoming to researchers outside mathematics who have genuine mathematics questions, without necessarily bouncing them off M.SE first. Mathematics has a history of both renewal from other fields and fruitful interchange, but often the initial interactions are fraught with communication problems. It is unfortunate that the question smells like homework, but that can be fixed with suitable effort by the poster.

(Incidentally, I have, in the past, wrongly voted to close interesting and nontrivial questions because I mistook them for elementary questions, so my hands are not particularly clean on this front.)

To direct such an OP to math.SE seems apt. They still can come back if the questions turns out to be 'too hard' for math.SE; possibly with an improved version. If OP is sure it is good to avoid this, they should read 'How to ask?' and follow it.

Your ininuation that in the absence of abatkai's comment the question would not have been closed seems a bit unlikley; it seems at least as likely to me that if it were not for that comment the question would have been closed without comment.

I will refrain, for the moment, to pursue various aspects and ramifications of this in detail. But I consider your behavior as rude.