A test paper has 4 multiple-choice questions, each with three options (A), (B), (C). Several students take the exam, and after grading, it is found that: any 3 students have 1 question where their answers are all different. How many students can take the exam at most?
Problem 664
Official solution
(提示: If 10 people take the exam, then for the 1st question, at least 7 people choose two options: for the 2nd question, at least 5 people among these 7 choose two options; for the 3rd question, at least 4 people among these 5 choose two options; for the 4th question, at least 3 people among these 4 choose two options. Therefore, these 3 people choose only two options for each question, which contradicts the problem statement. Thus, the maximum number of students taking the exam is 9. Additionally, it is easy to construct a scenario where 9 people take the exam.)