Please tell me which sentence is correct.

Adrian chose him and I for the job.

*Adrian chose him and me for the job.

If the asterisk before the second sentence is saying that you think the second sentence is correct, you're right! Nice job! =)

How would you explain this to a teacher that says the first one is correct?

Explain it like this:

"Would you say, 'Adrian chose I ...' or 'Adrian chose me ...'? The correct answer is "me" because it's a direct object. "I" is used only when the pronoun is the subject of a clause.

I think I would let the Principal explain it to the teacher.

Adrian chose him and I for the job.

The correct sentence is "Adrian chose him and me for the job."

To determine whether to use "me" or "I" in a sentence like this, you can remove the other person and see which pronoun sounds better. In this case, if we remove "him" and say "Adrian chose me for the job," it sounds correct. On the other hand, if we remove "me" and say "Adrian chose I for the job," it sounds incorrect.

Therefore, "Adrian chose him and me for the job" is the correct sentence.