Regular infinitive or bare infinitive after but?

Is there any rule for when to use to and when to drop it in expressions that include but (to) do?

A1-- You have no choice but to study hard.

B1-- You have no choice but study hard.

A2 -- I can't help but keep asking questions.

B2 -- I can't help but to keep asking questions.

A3 -- She could do nothing but give in.

B3 -- She could do nothing but to give in.

I guess all A's are correct. What about the B's? If they are incorrect, is it because of a rule, some logic, or because of idiomatic usage?

Infinitives after but meaning except are normally used without to.

She could do nothing but give in.

After cannot(can't) (help) but+ bare infinitive.

I can't help but keep asking questions.

We use to-infinitives after no choice, alternative, option but

You have no choice but to study hard.