grammaticality - Which are right choices in: “Can you imagine him/his forgetting his own birthday?”

Which one of the following is correct?

1. Can you imagine his forgetting his own birthday?


2. Can you imagine him forgetting his own birthday?

The question was asked in SNAP 2009 and I can't understand why they said that the only right answer is option 1.

I hear people saying 2 all the time, and I read it that way often enough, so what’s supposed to be wrong with option 2?

Or is the test itself wrong?