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

Answer

To solve the problem, we need to determine the maximum number of students in a class who can answer a set of five true-or-false questions, given that no student has answered all questions correctly and no two students have provided the same sequence of answers.

### Step-by-Step Solution:

1. **Understanding the Total Possible Answers**:
Each question has two possible answers: True (T) or False (F). For five questions, the total number of possible answer combinations is calculated as:
\[ 2^5 = 32 \]
This means there are 32 different ways to answer the five questions.

2. **Identifying the Correct Answer Sequence**:
Among these 32 combinations, there is exactly one sequence that represents all correct answers (let's denote it as C).

3. **Excluding the All-Correct Answer**:
Since no student has answered all questions correctly, we must exclude this one correct sequence from our total. Therefore, the number of sequences that can be used by students is:
\[ 32 - 1 = 31 \]
This means there are 31 different answer sequences that students can use, ensuring that none of them has the all-correct answer.

4. **Conclusion**:
Since each student must provide a unique sequence of answers, the maximum number of students who can participate in the class, given the conditions of the problem, is:
\[ \text{Maximum number of students} = 31 \]

### Final Answer:
The maximum number of students in the class is **31**.

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