I still think that as it is, this is not a good question. Motivation and context should be provided, then it may suit. This is what I tried to convey, maybe badly formulated. As it stands, it is a (maybe) hard exercise.

You should address people who voted to close... :-) I only expressed my opinion. I missed motivation and context. It was not clear how the question connects to research. There was one a consensus that hard and tricky elementary questions are borderline here, but of course this may change. We also do not discuss contest problems from The International Mathematical Olympiad if there is no connection to further research somehow.

I notice that you haven't answered many/any questions in this area. This makes me distrust your refusal to elaborate on how you are so sure this question is not research level. I think you should be more careful before warning people that a problem is elementary to make sure that the problem actually is elementary, and not just stated simply. I suspect that people trusted your assessment and closed the question without independently evaluating the problem.

As it stands, I can see a couple of awkward recurrence relations. I don't see how to get asymptotics or good estimates over the whole range of parameters, particularly where inclusion-exclusion leads to an alternating sum of large terms. If there really is an elementary solution, I would like to see it. However, the question is closed, so no one can post answers.

That a question is imperfect does not mean the question should be dismissed, that the author should be directed to leave the site, and that people should be blocked from answering it. Next time, how about asking for the context instead of declaring that the problem is off-topic?

It's no surprise that you disagree with me again, quid. I wish I had the time and energy you waste arguing past people.

I think calling a reasonable question elementary without due consideration is unacceptable. I'm running out of polite ways to say it.

I'm pretty sure the consensus we established here is that questions (that aren't obviously spam) should not be closed without comment. In other words, without András Bátkai's comment, we should have expected another comment. In some alternate universe (perhaps with low probability), that other comment would address the lack of context instead of the level, e.g. "please edit your question to include some motivation". At this moment, there is nothing stopping him from adding a clarifying comment or a retraction.

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.)

@Scott Carnahan: I am well aware of the in principle existing consensus (no close without comment) still it does not happen very infrequently (though not often) that questions (not actual spam) are closed without any comment. But yes the most likely thing should be the question being closed with a different comment, which could have been better or worse or about the same. (While initially not optimal, as acknowledged/clarified by abatkai, the one that was present is IMO rather to the good end since it is neutral to friendly. Unlike say "Do it yourself!", which one can also get on this site; seen various times, and just some 12 hours ago with 4 upvotes. [The question was addmittedly worse, still.])

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 also disagree with closing http://mathoverflow.net/questions/130319/combinatorics-show-the-formula-always-gives-an-integer-closed for similar reasons. Again, there was an early comment suggesting that the problem is a simple exercise, when it is not. I posted my answer when there were 4 votes to close. IIRC, all 4 of the votes were to close as "off-topic."

Again, I'm not pretending that the question was perfect. It suggests that there is a simple counting argument, when no such argument is known. I think the problem is nevertheless on-topic, as shown by the related papers by Ira Gessel and others.

1. In my observations, it is only seemingly elementary questions in probability and combinatorics where closing votes are justified by "lack of context" or "lack of motivation." No one seems to claim lack of context or motivations justifies closing votes or dismissive comments about questions about high-powered mathematical machinery.

2. Yes, most likely a few counterexamples to 1) exist. I'm sure some members of this forum can dig them up.

Nevertheless, even a a cursory observations reveals a clear pattern: questions about abstract nonsense that clearly came from professional mathematicians are usually answered, context or not, but questions about easy-to-understand probabilistic and combinatorial questions are often closed as too elementary.

Once it is pointed out, usually here on meta, that no plausible solution to the closed elementary-looking is known, many members of the community fall back on the "lack of context" or "poorly motivated" justifications.

3. It is difficult for me to take these reasons at face value. For example, any professional mathematician should be able to appreciate the motivation in desiring to understand
why a ratio of two integers turns out to be integer (or, alternatively, to see a proof of this statement which finds a transparent `explanation' - something not based on induction, for example).

4. In the discussions that ensue after the closing, apparently difficult combinatorial questions are often characterized as mere "exercises" (this appears in this very thread, in fact) apparently not worthy of research.

In light of that, it is difficult to resist the conclusion that the closing of these questions is at least in part a manifestation of the disdain many in the mathematical community feel towards combinatorics (i.e., "slums of topology" and all that - see also http://mathoverflow.net/questions/17226/is-discrete-mathematics-mainstream for a discussion of more modern attitudes).

5. 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.

6. There are some users here which spend an inordinate amount of time arguing back and forth on this board, usually being very careful to misunderstand the arguments of their interlocuters. For this reason, I regret that I will not be able to participate in extended discussions here.

I wouldn't read too much significance into the case of the (2m)!(2n)!/(m!n!(m+n)!) problem, since this question has some unusual attributes. For one thing, it was clearly stated as a problem the OP had read somewhere, with a (presumably incorrect) memory that the source said there's a "simple counting argument". I have sympathy for the OP and am glad it was answered on MO, since it's incredibly frustrating when when you've been told there's a combinatorial proof but you just can't find one at all. However, I see the misinformation about a combinatorial proof as being the primary argument for why it's appropriate for MO. If you are satisfied with any proof of integrality, then this problem is a common and not too difficult exercise in discrete math or combinatorics courses, so if someone just posted "Show that (2m)!(2n)!/(m!n!(m+n)!) is an integer" I would vote to close. Furthermore, if there really were a simple counting argument that proved it, then that would presumably make it even less appropriate for MO. It's too bad the question got closed, but given that it basically stated "here's a problem I read somewhere that has a simple solution; how do I solve it?" I'm not surprised it got closed at first.

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.)