For a set of four multiple choice questions with three options (multiple correct possible), no two students have given the same sequence of answers. What is the maximum number of students that sat for the test, for this to be possible, given that each student attempts all the questions. No student gives all the correct answers?

Options

A. 4095
B. 2400
C. 1295
D. 2500

Answer

To solve the problem, we need to determine the maximum number of students that can answer a set of four multiple-choice questions with three options (A, B, and C) under the condition that no two students have the same sequence of answers and that no student gives all the correct answers.

### Step-by-Step Solution:

1. **Understanding the Choices for Each Question:**
Each question has three possible answers: A, B, or C. A student can either choose one of these options or leave the question unanswered. This gives us two choices for each option (to choose or not to choose).

2. **Calculating the Total Choices for One Question:**
For each question, the choices can be represented as:
- Choose A
- Choose B
- Choose C
- Leave it blank (not choosing any option)

Therefore, for each question, there are \(3\) options (A, B, C) and \(1\) option for leaving it blank, leading to a total of \(4\) choices per question.

3. **Calculating Total Combinations for All Questions:**
Since there are \(4\) questions, and each question has \(4\) choices, the total number of combinations of answers for all questions is:
\[ 4^4 = 256 \]

4. **Excluding the Case of All Correct Answers:**
However, we need to exclude the case where a student answers all questions correctly. There is exactly \(1\) way to answer all questions correctly (choosing the correct option for each question). Thus, we subtract this from the total combinations:
\[ 256 - 1 = 255 \]

5. **Conclusion:**
Therefore, the maximum number of students that can sit for the test, ensuring that no two students have the same sequence of answers and that no student gives all correct answers, is:
\[ \boxed{255} \]