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

Solution 1:

Traditional grammar insisted that since forgetting in your example was a gerund, i.e. a present participle used as a noun, then it could only be specified by a noun/pronoun in the possessive case. Further, it was assumed that a gerund, unlike participles, could not take subjects.

If you apply this rule, which the test assumes you have learned, then you will choose his over him. Many writing guides, even as YouTube videos, continue to insist on this rule.

While this construction is still quite frequent in more formal written language, it is quite rare in unrehearsed spoken English except among older speakers who have incorporated the rule into their own idiolect. The trend since the 1950s, however, selectively aligns standards for writing with those for educated speech, so that today, many writers would consider using the possessive a relict of the 19th century.

Insisting on the possessive in your example, unless the makers of the test are close to retirement age, is simply pedantic. A simple remark about someone forgetting his own birthday is hardly in a formal register, so the objective case would be the more idiomatic choice.