A disclaimer: I’m a mathematician, so I hope I’m not confusing the matter by speaking too generally. If I do, please tell me so I can clarify. I don’t bite!

I think part of the problem here is in the way you phrased the question.

Some terminology:

A mathematical **expression** is a sequence of numbers (or variables, but we won’t go there) connected by operations, something like ”(2+4–8)*7”.

An **equation** is two expressions with an equals sign in between them, like “3+7=10”.

What it sounds like you’re asking is “Once I fix a given number, how many expressions can I write down that are equal to that fixed number?” If that’s not what you’re asking, please tell me, as I have misunderstood you.

The answer to this question, as @hannahsugs has explained, is that there are infinitely many such expressions. But since that’s not a particularly interesting answer, a mathematician’s instinct is to modify the question slightly and see if you get a more interesting result. If your goal is to impose restrictions on the original problem in order to get a finite answer, there are several ways to go about it.

First, you have to determine how you’re going to count these expressions. While this may seem obvious (“I learned to count in kindergarten, thank you very much”), it actually presents some real difficulties. For example, if you count “3+4”, “3+4+0”, “3+4+0+0”, etcetera, as different expressions, then there is no hope of getting a finite number of possibilities. (Do you see why?) The same is true of multiplication by 1.

Additionally, in order to have any hope of getting a finite number of combinations, you have to disallow any pair of operations that can “cancel” each other. This is because if you allow both addition and subtraction, you can “add zero” without making it look like you’re adding zero. For example, 3+4+9–9=3+4+0, so I can write down something like 3+4+9–9+9–9+9–9… for as long as I want, and it will still equal seven. I can even be sneaky and re-order things, and write something like 3–9-9+4+9+9. The same is true if you allow both multiplication and division (do you see why?)

So in order to get a finite number of expressions, we can’t allow addition of zero, or multiplication by 1, and either allow addition or subtraction, but not both, and either multiplication or division, but not both.

However, addition and subtraction are really the same thing, since 3–2 is the same as 3+(-2). Similarly, multiplication and division are the same, since 2/4 is the same as 2x(¼), or 2x(.25). So what I really mean by not allowing both addition and subtraction or both multiplication and division is that you have to restrict the kinds of numbers you can add or multiply.

The most natural way to do this is to only allow yourself to work with addition and multiplication of positive whole numbers (with the proviso that if you’re multiplying you have to be multiplying by something bigger than 1). That way, whenever you do an additional operation, you end up with something bigger than when you started, so once you reach your target number, you stop. In this case, there will be a finite (although possibly very very large) number of expressions that equal your target number.

Are you asking about the nature of infinity? About what makes some things finite and some infinite? Because that’s a whole other can of worms, and a topic for another day.