Which one is correct?

Which one is correct among the next sentences? If there are more than one correct sentences, teach me the differences, please.

Both teacher and student agreed to reschedule the session. correct

Both the teacher and the student agreed to reschedule the session. (This sounds better if you are talking about a specific teacher and student.)

Both of teacher and student agreed to reschedule the session. incorrect

Both teacher and student agreed with rescheduling the session. incorrect

Both teacher and student agreed with the rescheduling the session.

(With this sentence, 'rescheduling the session' is what the student and teacher agreed on, so you need 'the' before 'rescheduling' to make it correct.)

Both of teacher and student agreed with rescheduling the session. incorrect

Which one is correct?

Which one is correct among the next sentences? If there are more than one correct sentences, teach me the differences, please.

Both teacher and student agreed to reschedule the session. (correct)
Both of teacher and student agreed to reschedule the session. (incorrect)
Both teacher and student agreed with rescheduling the session. (correct)
Both of teacher and student agreed with rescheduling the session. (incorrect)

The first sentence is correct, however it would make more scence to say "Both the teacher and the student agreed to reschedule the session."

The third sentence is correct but not a normal way of saying it, instead you would say "Both the teacher and the student agreed with rescheduling the session."

The difference between the first and third sentences is that the first implies that the teacher or the student has suggested the rescheduling, whereas in the third sentence, it implies that somebody else has made the suggestion.

The second and fourth sentences do not make sence because there is not more than one teacher or more than one student in the situation, meaning you do not need "of". For example, if there was one teacher and two students, it would be correct to say "The teacher and both of the students agreed to reschedule the session".

Which one is correct?

Which one is correct among the next sentences? If there are more than one correct sentences, teach me the differences, please.

Both teacher and student agreed to reschedule the session. Correct
Both of teacher and student agreed to reschedule the session. Incorrect
Both teacher and student agreed with rescheduling the session. Incorrect
Both of teacher and student agreed with rescheduling the session. Incorrect

The third one could be - Both teacher and student agreed to the rescheduling of the session.