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: Understanding the Question: There are five true/false questions. Each student answers with a sequence of five answers (each either true or false). No student has the all‑correct sequence, and no two students have the same sequence. We need the maximum possible number of students.

Step 2: Key Formula or Approach: Total possible distinct answer sequences = \(2^5 = 32\) (each question has 2 choices). Exclude the one sequence that is all correct.

Step 3: Detailed Explanation: Number of all possible answer sequences = \(2^5 = 32\). If we remove the single sequence that is entirely correct, we are left with \(32 - 1 = 31\) sequences. Since no two students can have the same sequence, the maximum number of students is exactly the number of distinct sequences available, i.e., 31.

Step 4: Final Answer:
Maximum number of students = 31, which corresponds to option (B).