From an article of The Daily Mail (2007):
[...] first immigrants from Jamaica [...]
There should have been no question about their being admitted
Why does one/the text use their instead of them?
If you google for "no question about you being", you find 3.1 million search results, whereas "no quesiton about your being" only yields 849.000.
Are both forms correct?
Both forms are correct. One way or another, something has to be the object of the preposition "about".
The word "being" can be either the gerund form or a participial form. Both gerunds and participles take arguments, so "being accepted" can be either a gerund phrase or a participial phrase.
If it is a gerund phrase, then it can act on its own as the object of the preposition "about', which means that "about being accepted" makes sense. The gerund phrase can be modified by the genitive pronoun "their", which means that "about their being accepted" makes sense.
If it is a participial phrase, then it does not act on its own as the object of "about". However, the objective pronoun "them" can. The phrase "about them" makes sense. Participial phrases can modify nouns and pronouns, which means that "about them being accepted" makes sense.
Both forms are correct, but both forms are not applicable in every situation. In the sentence you quote, the question at hand is about the admission, not about the people. "Being admitted" is the more sensible object for the preposition in this case.
"being admitted " is considered by grammarians to function as a noun-phrase:
My being admitted to the club was cause to celebrate.
A pronoun in one of the objective cases (accusative [direct object], dative [ indirect object]) cannot (technically speaking) modify a noun-phrase.
[COLLOQUIAL] Me being admitted to the club was cause to celebrate.
But native speakers don't pay attention to such rules, and say things like:
There should have been no question about them being admitted.
Whether they do so because them comes after *about" (about them, which makes sense), or because "them" is understood to be a kind of demonstrative pronoun in the nominative, and is the subject of a pseudo-verb "being", is open to debate.
