A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.
Problem
Source: Chinese Mathematical Olympiad 2000 Problem 6
Tags: combinatorics unsolved, combinatorics, Set systems