Question

For a set of 5 true/false questions, no student has written all correct answers 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?

Answer 1

First, list all the possible sets of answers that can be made from these 5 true/false questions. There will be only one set of answers that will be correct among all the sets of the answer. As no student has written all correct answers, then subtract the correct set of answers from all possible sets of the answer. Also, no two students have given the same sequence of answers. So, the remaining sets of answers will give the number of students in the class.

Complete step-by-step answer:
Given:- A set of 5 true/false questions. No student has written all correct answers and no students have given the same sequence of answers.
As every question has 2 options either true or false and there are a total of 5 questions. So, the total number of sets of the answer will be,
${2^5} = 32$
There will be only one set of answers which will be correct.
So, the number of sets of incorrect answers is $32 - 1 = 31$.
As no student has written all the correct answers. It means the number of students in the class must be less than or equal to 31.
Also, no students have given the same sequence of answers. It means all students have given wrong and different sets of answers.

Hence, the maximum number of students in the class for this to be possible is 31.

Note: The fundamental principle of counting is used to find the total ways.
The fundamental counting principle is a rule used to count the total number of possible outcomes in a situation. It states that if there are $n$ ways of doing something, and $m$ ways of doing another thing after that, then there are $n \times m$ ways to perform both of these actions. In other words, when choosing an option for $n$ and an option for $m$, there are $n \times m$ different ways to do both actions.