**Question:**

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?

**Solution:**

Given: a set of five true – false questions.

To find: the maximum number of students in the class.

Condition: no student has written the all correct answer and no two students have given the same sequence of answers.

The total number of answering a set of 5 true or false question $=2^{5}=32$

Since, no two students have given the same sequence of answers and no student has written the all correct answer.

Therefore total possibilities reduces by 1 (of no student has written the all correct answer)

$\Rightarrow 2^{5}-1=32-1=31$