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I was wondering which of the following phrases are correct?

"without their having to learn" or "without them having to learn"?

Any help is highly appreciated.

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Where a gerund-participle clause is complement to a preposition, both genitive and non-genitive subjects are possible:

I have no objections to [their/them taking notes].

She insisted on [my/me being present at the interview].

So, both your examples are fine. It's essentially a free choice between genitive "their" and non-genitive "them", though the genitive is fairly formal compared to the non-genitive.

Note that if a non-genitive subject is a personal pronoun, it always takes the accusative case ("me/him/her/us/them").

  • Would you please explain your last paragraph a little more. Thanks. – user2521204 May 26 '16 at 10:58
  • Sure: if the subject is a personal pronoun, it can be the genitive "my/your/his/her/our/your/their", or the accusative "me/him/her/us/them", but not the nominative "I/he/she/we/they". – BillJ May 26 '16 at 16:34

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