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For example:

  1. We need to be able to tell the differences between each kind of problem.
  2. We need to be able to tell the differences between each kinds of problem.
  3. We need to be able to tell the differences between each kind of problems.
  4. We need to be able to tell the differences between each kinds of problems.

What rule in English are we dealing with here? (2) and (3) sounds off, but I can't explain the grammatical reasoning behind it.

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difference can be countable or uncountable. When we use it in the set phrase tell the difference, we almost always treat it as uncountable, so we use difference in the singular, as this NGram graph shows. Here is the correct sentence:

We need to be able to tell the difference between ...

If you want to make it clear that there are multiple differences between any two kinds of problem, it would be better to use spot or identify, because these are not associated with the uncountable form of difference.

We need to be able to spot the differences between ...

When you use between, you have to follow it by at least two things:

What is the difference between a cat? - incorrect
What is the difference between a cat and a dog? - correct

each always describes just one item out of many, so the noun that follows it must be singular. Remember, though, that between has to be followed by at least two things, so you can't use each kind after between.

If, instead, you use different or the various, that means that you are talking about at least two things, so you use plural, and it works after between:

We need to be able to tell the difference between different kinds of problem

When you use the word of, you generally follow it by a singular.

... different kinds of problem

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