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The fact that John was a good student resulted in his/him having many offers for the job.

Which one should be used and why?

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Both are correct, but the possessive form is being used less and less these days.

If you take a look at this Google Ngram you can see how the use of him saying vs his saying evolved over time.

That being said, some English teachers often still consider the non-possessive form as incorrect, though I think it is widely accepted nowadays. So to answer your question: both options can be used and they are both correct.

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