For a set of five true or false questions, no student has written the all correct answer and no two students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?

Hint: We will first start by finding the all ways of answering 5 questions by finding the way in which each question can be answered and then using the multiplication principle to find the total ways. Then we will subtract the case in which all answers are correct to find the final answer.

Complete step-by-step answer:
Now, we have been given a set of five questions with true or false answers. Also, it has been given that all the students have written different answers and no two students have given the same sequence of answers. Also, no student has written the correct answer.
Now, we know that each question has 2 options, therefore, 2 ways of answering a question. Similarly, each question out of five has 2 ways of answering. Now, from the fundamental principle of counting we have the total ways in which 5 questions can be answered as,
\[2\times 2\times 2\times 2\times 2=32\]
Now, in 32 ways there will be 1 way that gives all correct answers but it has been given that no one has given all the answers correctly. So, we have the total different ways of answering questions as $32-1=31$.
Now, since none of the students has the same sequence of answers, therefore, there can be at most 31 students in the class.

Note: To solve these types of questions it is important to note that we have used the fundamental principle of counting to find the total ways. Also, it is important to notice how we have first found the different sequence of answers possible and used it to find the number of students.