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I am planning to ask a question which requires you to count the number of possible ways to arrange something with certain conditions. Are these on-topic?

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Yes, is on-topic.

Counting problems are essentially combinatorics, and this is just as much on-topic as other kinds of mathematics. You can browse the linked tag to see what kind of questions we've had on this topic.

As Glorfindel points out, there are caveats to what kinds of maths puzzles can be on-topic. Basically if it's interesting/neat/challenging enough to count as a puzzle, then it's fine, but if it's just a basic/rote problem, then it's likely to be closed.

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Let me quote this high-scoring answer by @xnor on Are math-textbook-style problems on topic?:

So, what makes something a math puzzle rather than math problem? I think there's a few features.

  • Clever or elegant solution, often an "aha" moment
  • Unexpected problem statement.
  • Unexpected or counterintuitive result.

Counting problems are math problems, so I think this advice applies here as well. The first and third point can apply to your problem, but you need to know the solution upfront and judge whether those apply or not.

There is a recent counting problem question on our sister site Mathematics Stack Exchange which (IMHO) could be rewritten as a suitable puzzle for us: Number of ways to stack LEGO bricks. Incidentally, it could also qualify for the second point, though that is rather subjective.

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