Question

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

• A. 30
• B. 31
• C. 32
• D. 16

Hint:

For \(n\) true/false questions, total sequences = \(2^n\). If one particular sequence is forbidden, the maximum number of distinct responses is \(2^n - 1\).

Answer 1

The Correct Option is B

Step 1: Understand the setup.
There are five true-or-false questions. Each student gives one answer sequence. We want the largest possible number of students so that all sequences are different and none of them is the fully correct sequence.
Step 2: Count all possible sequences.
Each question has 2 choices, True or False. With five questions the number of different answer sequences is \[ 2\times2\times2\times2\times2 = 2^5 = 32. \]
Step 3: Use the no-two-same condition.
Since no two students share a sequence, each student must use a different one of these 32 sequences. So at most 32 students.
Step 4: Use the no-all-correct condition.
Exactly one of those 32 sequences is the fully correct one. We are told no student wrote all correct answers, so that one sequence cannot be used.
Step 5: Remove the forbidden sequence.
Taking away that single all-correct sequence leaves \[ 32 - 1 = 31 \] usable sequences.
Step 6: State the maximum.
So at most 31 students can each have a distinct sequence with none being all correct. \[ \boxed{31\ \text{students}} \]