Question

In a test, there are 7 questions of the type 'True or False'. No student got all the answers correct. If the sequence of answers for every student is unique, then the maximum number of students who appeared for the test is \(\ldots\)

• A. \(128\)
• B. \(126\)
• C. \(127\)
• D. \(125\)

Hint:

Total sequences 2^7 minus the one all-correct sequence.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Each of the 7 True/False questions can be answered in exactly 2 ways. Every student has a different sequence of answers, so the number of students cannot exceed the number of possible sequences.

Step 2: Key Formula or Approach:
By the fundamental counting principle, the number of answer sequences is \(2^7\). One of them is the all-correct sequence, which no student has.

Step 3: Detailed Explanation:
Total sequences:
\[ 2^7 = 128 \]
Remove the all-correct sequence, since no student got all answers correct:
\[ 128 - 1 = 127 \]
Because every student has a unique sequence, the maximum number of students equals the number of allowed sequences, 127.
Option (A) 128 includes the forbidden all-correct sequence. Options (B) 126 and (D) 125 remove more sequences than the one excluded.

Final Answer:
The maximum number of students is 127, option (C). \[ \boxed{127 \text{ (C)}} \]