The correct option is A None
The sentence means that no student could solve the problem, which is why the teacher solved that problem. All the given options are indefinite pronouns, and we have to choose the most suitable one. Indefinite pronouns refer to a person or thing in general, and do not direct toward a particular person or thing. 'None' means 'no person', 'not part of the whole', and when placed in the sentence, 'none of the' means, 'not one student from the whole group of students', the reason why the teacher solved it. Hence option D is correct. 'Someone' and 'somebody' refer to an unspecified person, both suggest a presence of a person, whereas the sentence calls for an absence or negation of a student. They're both singular, and do not refer to a person in terms of a group, whereas the sentence needs a pronoun that refers to a person in terms of their group, hence options A and D are incorrect. 'Anybody' also refers to a person, it means 'doesn't matter which person', but the sentence wants a pronoun that refers to 'no person', hence option B is incorrect.