For a set of five true-or-false questions, each student gives a sequence of answers. If no student writes the completely correct sequence and no two students give the same sequence, what is the maximum possible number of students?

Each question has 2 possible answers, so the total number of answer sequences is $2^5=32$. Exactly one of these is the all-correct sequence, which is forbidden. That leaves $32-1=31$ allowable distinct sequences, so at most 31 students can all have different answer patterns.