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Oct 24, 2009 at 21:56 comment added Michael Lugo I think this answer is reasonable. But this implicitly requires that permutations be one of the first objects you look at. As Qiaochu has pointed out, we don't necessarily have to make this choice.
Oct 24, 2009 at 21:32 comment added Qiaochu Yuan Combinatorics doesn't really fall under the purvey of this question, since it's both relatively non-axiomatic and highly non-linear. And for what it's worth, I consider inclusion-exclusion highly nontrivial, at least conceptually (as Mobius inversion).
Oct 24, 2009 at 21:30 comment added Harrison Brown I think I would disagree, since counting derangements is just a standard application of inclusion-exclusion, which is pretty trivial.
Oct 24, 2009 at 21:08 comment added Jonah Ostroff I'm not at all convinced that this is the best answer, but after a few minutes of scratching my head, I turned to an old undergraduate textbook. Six chapters in, this is the first proven result that seems to qualify.
Oct 24, 2009 at 21:07 history answered Jonah Ostroff CC BY-SA 2.5