Hi aaba,
Thank you for your quetsion. This one was trickly, so I took some time to look over this question, and I think I have a good strategy for answering it quickly.
One of the first things that jumped out at me when looking over the answers quickly was pronoun usage. So let's look over each answer, checking to make sure there are no vague pronouns or pronoun agreement issues:
A: that a university permits
their failing students who would otherwise abandon postgraduate education to pursue
This is a
WRONG answer because the pronoun "their" is too vague. It's unclear whether that's referring to the professors' students or the university's students.
B:
universities to permit
their failing students who would otherwise abandon postgraduate education that they can pursue
Since we're talking about a group of professors proposing one plan for failing students, it's more likely that the professors all work at the same university. Therefore, this is
WRONG because it messes with the intended meaning.
C: that a university permit
its failing students who would otherwise abandon postgraduate education to pursue
This is
CORRECT because the pronoun "its" is clearly referring to "a university," which is a collective noun that always gets a singular pronoun. There is no mistaking "its" as referring to the plural "professors."
D: universities permitting
a failing student whose postgraduate education would otherwise be abandoned the pursuit of
This is
WRONG because it uses the singular phrase "a failing student." This doesn't agree with the later plural phrase "until
they improve their grades."
E: universities to permit
a failing student whose postgraduate education would otherwise be abandoned the pursuit of
This is
WRONG for the same reason as D - the use of the singular "a failing student" doesn't agree in number with the plural "until they improve their grades."
So, by looking at pronoun agreement and vague pronouns alone, we can easily narrow it down to one correct answer!