There are even numbers of good and bad things.
There are an even number of good and bad things.
What is the difference?
There are even numbers of good and bad things.
There are an even number of good and bad things.
What is the difference?
There are even numbers of good and bad things.
That implies, to me, that there are two sets - whether the sets are composed jointly or separately of (a) good, (b) bad is unclear - and that both sets are composed of an even number of contents.
There are an even number of good and bad things.
That implies, to me, that there is one set, composed of good and bad things, and that the number of them, in total, is even, but...
I cannot agree with the comments above that there is no difference in meaning. The second sentence, without the 'an' would read:
There are even number of good and bad things.
The lack of concord between verb and subject makes this meaningless.
It does, however, raise the point that the use of 'number' in the singular and 'numbers' in the plural in the above examples is ambiguous.
You could get around it by rephrasing to:
There is an even number of good and bad things
(One set, mixed)
Or
There are even numbers of good things and (of) bad things
(Two sets)
The question then arises whether you are implying equality and balance in your use of even (as in 'We're even now' in a tit-for-tat scenario) or whether you simply mean 'not odd'.
For the meanings, they are highly ambiguous. I can think of many possible meanings:
There are good and bad things. The number (count) of good things is even (not odd), and the number (count) of bad things is even (not odd).
The sentence express the shock of the author that the "numbers of good and bad things" are actually in the place where the author did not expect.
There are [a number of] good things and there are [a number of] bad things. Collectively, the number (that is, the sum) is an even number (as compared to an odd number).
The number of good things is the same as the number of bad things. That is, they (the numbers) are even (as in level)
For meaning number 1, since there are two numbers being referred to in the meaning, then it must be the case that the sentence use the plural form of "numbers" and this is the case only for the first sentence. So I will assign meaning number 1 to the first sentence, and it parses as:
There are [even numbers] of [good and bad things]
Analogous to:
There are [huge bags] of [candies and chocolates]
In the sense that there are bags containing candies ans chocolates, and each of them is huge.
Meaning number 2 (also applicable only to first sentence), although syntactically plausible, but it is semantically odd. So I guess this is not what the sentence is intended to mean. But for completeness, it is parsed as:
There are [even] numbers of good and bad things.
Analogous to:
That list is so comprehensive. There are [even] numbers of global population from 20 years ago!"
For meaning number 3, since there is actually only one number (that is, the sum), then it is only applicable to second sentence and in which case the sentence is phrased as:
There are [an even number of] [good and bad things]
Analogous to:
There are [six] pens and pencils.
For meaning number 4, since the two numbers here are to be claimed the same, they can be in plural form or singular form. However, since the object here is the "number", the verb must also be singular, so it doesn't match second sentence. The first sentence is parsed as:
There are [even [numbers of good and bad things]]
Analogous to:
There are [different [numbers of good and bad things]]
And if we were to use the singular form I would use:
There is [an even [number of good and bad things]]
Okay, but those are very thorough analysis of the sentences and meanings, which rarely happen if you just use your sentence verbally. In those cases, take the first meaning that comes to your mind. For me, I will get meaning number 4, and for the second sentence, I will actually consider the sentence to be odd, and I will change it in my mind to the singular case of meaning number 4.
So if you tell me either sentence during conversation, I will understand it as you're claiming that there are as many good things as there are bad things.
To me the first suggests that there are lot of things which can be categorised as good or bad, and, overall, the numbers even themselves out.
The second suggests everything has been categorised as good or bad, and there is a perfect 50/50 split.
Not sure why, that's just how it reads to me.