Plural or single with compound objects

Suppose you want to say something like

There are two crates, having three and one balls respectively.

How to say this correctly?

Is it ‘…one balls’ or ‘three balls and one ball’, or something else?

Solution 1:

As with a comment under the question, I'm not sure about the overall construction of the sentence, but I will focus on the actual question first:

There are two crates, having three balls and one ball, respectively.

That is the full version version of the sentence.

The problem you're having is how your shortening it. Regardless of actual grammar, neither of these sounds right:

✘ three and one balls ✘ three and one ball

There are two ways of getting around it. One is to provide the longer phrase, as you suggested. The other is to reverse the position of the numbers:

✔ There are two crates, having three balls and one ball, respectively.
✔ There are two crates, having one and three balls, respectively.

The formation of the second variation matches some similar situations.

Subject-Verb

✘ Your pens and notebook are out of date.
✔ Your notebook and pens are out of date.

Although the combined subject is plural (and, in context, is grammatical), the plural verb only sounds natural if it comes immediately after a plural subject.

Ellipsis

I am eating an apple and cherry.

That sentence is correct. But based on ellipsis, if we were to literally apply the written article to each of the items, we would get this:

✘ I am eating an apple and an cherry.

But we don't do that. Along with being able to assume the missing article in front of the second item, we also assume the correct article:

✔ I am eating an apple and a cherry.

Applying both of the above to the second variation of your sentence, the following is observed in terms of subject-verb agreement and ellipsis:

There are two crates, having one (ball) and three balls, respectively.

Finally, although that answers your actual question, I believe the sentence itself would read better if it had a different construction.

Using the second variation I mentioned to get around the problem you are having, I suggest restating it in one of these ways:

There are two crates, containing one and three balls, respectively.
There are two crates that have one and three balls, respectively.
There are two crates that contain one and three balls, respectively.
(The / these / those) two crates have one and three balls, respectively.
(The / these / those) two crates contain one and three balls, respectively.