Question

At a university all students must take a test to see if they understand campus polices. The test has 6 multiple choice questions, each with three possible answers and only one right answer per question. This year every student got at least two questions right. How many possible ways could the students have answered the test questions?

The incoming class this year has 541 students. Prove that at least two students answered the test in the exact same way.

Answer 1

The students got at least 2 questions correct. So, in terms of correct and wrong, there are possible ways to answer. First we can pick two questions in ways and numbering of these question matters, so number of ways to choose two correct answered questions is . Now, for the rest of four questions, thy may have answreed it either correctly or wrongly. So in total, 480 possible ways to answer the questions.

Now, there are a total of 541 students, so by pigeon hole principle, at least students must have given the same answer. Here denotes the greatest intger equal or lesser than n. So, at least 2 students have given the same answer.