Question

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct answers. If every student has attempted all the questions, then the maximum possible number of students who have written the test is:

• A. \(80\)
• B. \(63\)
• C. \(15\)
• D. \(11\)

Hint:

When dealing with combinatorial problems involving restrictions, always consider the total possibilities first, then subtract the cases that violate the given restrictions.

Answer 1

The Correct Option is B

Step 1: Calculate the total number of different possible answers a student can give. 
- Since there are 3 questions each with 4 options, the total possible combinations are \(4^3 = 64\).
Step 2: Adjust for the constraint that no student can answer all questions correctly. 
- Subtract the one combination where all answers are correct, leaving \(64 - 1 = 63\) possible ways to answer.
Step 3: Given that no two students answer identically, the maximum number of students who could have written the test without any of them having all correct answers is therefore \(63\).