I read an IELTS test, which has the sentence:
Universities should accept equal numbers of male and female students in every subject.
At first, I thought it mentions two objects (male and female), so we have to use "the numbers of" but after reading some sample articles, I suddenly realized that some authors wrote:
Considering these, the idea of determining the number of males and females in every subject is justifiable.
So what's the difference?
In the example:
Considering these, the idea of determining the number of males and [the number of] females in every subject is justifiable.
the second the number of is omitted. Omissions commonly occur in parallel structures like this one. I am not excluding the possibility of considering the general number of persons regardless of the categories they are grouped in. But this context makes it apparent that the author of this sentence has in mind the number corresponding to each group. The accepted answer to the question: “Numbers” or “number?” Which sentence is correct? explains this as well.
In your first example there is clearly more than one number, each number corresponding to each group of elements, to each university and to each subject. Moreover, these numbers are compared to each other, they are equal. So it is perfectly logical to say :
Universities should accept equal numbers of male and female students in every subject.
There are two numbers here: the number of men and the number of women. These can also be called, “the numbers of men and women.” The parallel phrases can also be implicit, as in “the number of women and [the number of] men.”
Since two equal numbers are really the same number, you could also say, “the same number of men and women.” You might also hear the parallel construction, “an equal number of women and men.” I think it makes the most sense to analyze this as a reduction of “*an equal number of women and [an equal number of] men ...,” (analogous to, “a lower number of men and a higher number of women”) but we’d never actually expand the parallel items “an equal number ... and an equal number” that way.
Here are the basic rules:
Numbers of
You will use “numbers of” when referring to something in high quantities in the plural. This will accompany verb agreement where applicable. Consider the following examples:
There are huge numbers of people in the crowd.
When we arrived at the show, the numbers of people present were overwhelming.
Number of
Since “number of” suggests singular usage, you will apply it when there is only one noun in question. However, it will also suggest a kind of plurality. For instance:
I have told you a number of times that I don’t want to go camping.
No one can ever know the whole number of historical data in their lifetime.
Did you see the number of products they have at the fruit stand?
The number of crystalline salt grains is uncountable.
That second rule can be a bit tricky. Think of it like this: it's used when referring to a singular something involving multiple somethings.
I have told you a number of times that I don’t want to go camping. (Singular something: the act of telling you.)
No one can ever know the whole number of historical data in their lifetime. (Singular something: the non-discrete/uncountable reference to "historical data." See Note #1.)
Did you see the number of products they have at the fruit stand? (Singular something: the fruit stand. See Note #2.)
The number of crystalline salt grains is uncountable. (Singular something: the non-discrete/uncountable salt grains. See Note #1.)
So you're correct that it's important to identify the object being discussed. Here's your first example:
Universities should accept equal numbers of male and female students in every subject.
The object is plural, "students." The words male and female are adjectives modifying students. Now let's look at your second example.
Considering these, the idea of determining the number of males and females in every subject is justifiable.
What's being discussed? Idea. Perhaps more explicitly, it's the process of making the determination that's the object — and it's singular. The fact that the process involves analyzing something plural doesn't affect how we use the phrase "number of."
Note #1
In English, when we refer to something that's "too big to understand," (a non-discrete or uncountable object) we refer to it as a singular. The water in the ocean, the stars in the sky, sand on the beach, etc.
The number of gallons of water in the ocean.
The number of stars in the sky.
The number of particles of sand on the beach.
In each of these cases, we can't really know how much of the objects discussed exists. We might estimate it, but we can't really know. When this happens we treat the object as singular.
Note #2
I thought this was a curious example for the linked website to use. It's right, but for a very different reason than was presented in Note #1.
The number of products at the fruit stand.
The number of products in our warehouse.
The number of products on my desk.
The idea behind the word "products" isn't so great that it can't be counted (like "stars in the sky"). It's because the word "products" is one of those words that's just used that way. This issue is something similar to using the words "deer" and "sheep" as their own plurals. Why do we do that? Well... because we do. The word "products" is intrinsically uncountable.
The number of deer in the forest.
The number of sheep in the field.
And if you think that's funny, using the word "product" in the same way as "deer" and "sheep" is becoming more popular.
she uses product in her hair.
We shipped a lot of product this month.
Language changes... and this is a particularly good example of that change.
