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I have a task to translate rather large book in combinatorics into English. As it usually happens with math textbooks, there are lots of exercises, and thus, many long questions. I feel that the canonic word order is not acceptable in many of them as it confuses the reader. However, I am not sure about the best way to construct such questions. Example:

How many different integer-valued triangles are there, with the perimeter of 30?

How many different integer-valued triangles are there, having the perimeter of 30?

These questions are not too long but they illustrate my point well. My questions are:

  1. Is it acceptable to put "are there" in that part of a question?

  2. Is it acceptable to use "with"/"where"/"having" in such positions in a question?

  3. Should I put comma in front of "with"/"where"/"having"?

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I think you're correct: the "are there" part of those questions may not be verbose or ungrammatical, but it does read a bit clumsily.

I would suggest one of these versions:

How many different integer-valued triangles have a perimeter of 30?

How many different integer-valued triangles exist with a perimeter of 30?

If P is a triangle's perimeter, how many different integer-valued triangles exist where P = 30?

That last form would not likely be used in casual conversation, but it's perfectly fine in a math textbook.

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How many different integer-triangles [or integral triangles] are there (with | having) a perimeter of 30?

That is grammatical. This is also grammatical:

How many different integer-triangles (with | having) a perimeter of 30 are there?

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