Number of questions required to prepare a multiple choice quiz such that each student has different order of questions

If there were two questions on the quiz, we could prepare two quizzes with the questions in different order -- 2! = 2.
If there were three questions, we could get 3! = 6 different orders.
If there were four questions, we could get 4! = 24 different orders -- not quite enough for the class of 27 students.
If there were five questions, we could get 5! = 120 different orders - which is enough for 27 students.
The teacher will need at least 5 questions on the quiz.