Example 17 In an exam, there are 5 multiple-choice questions, each with 4 different answers to choose from, and each person selects exactly 1 answer per question. Among 2000 answer sheets, it is found that there exists a positive integer $n$, such that in any $n$ answer sheets, there are 4 sheets where the answers of any two sheets differ in at least 2 questions. Find the minimum value of $n$.
(2000 CMO Problem)
Solve: Let the 4 answers for each question be denoted as $1,2,3,4$, and the answers on each test paper be denoted as $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$, where $a_{i} \in\{1,2,3,4\}(i=1,2,3,4,5)$. Let the quadruples be
$$
\{(1, i, j, k, h),(2, i, j, k, h),(3, i, j, k, h),(4, i, j, k, h)\}(i, j, k, h \in\{1,2,3 \text {, }
$$
$4\}$, there are $4^{4}=256$ such quadruples. Since $2000=256 \times 7+208$, by the pigeonhole principle, there are 8 test papers whose answers belong to the same quadruple. After removing these 8 test papers, there are still 8 test papers in the remaining 1992 test papers that belong to the same quadruple; removing these 8 test papers, in the remaining 1984 test papers, there are again 8 test papers that belong to the same quadruple. Removing these 8 test papers, together with the previous two sets of 8 test papers, a total of 24 test papers are obtained. Among these 24 test papers, any 4 test papers always have 2 test papers whose answers belong to the same quadruple, and their answers to the last 4 questions are exactly the same, which does not meet the requirements of the problem. Therefore, the minimum value of $n$ is 25.
On the other hand, let
$$
S=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right) \mid a_{1}+a_{2}+a_{3}+a_{4}+a_{5} \equiv 0(\bmod 4), a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \in\{1,2,3 \text {, }\right.
$$
4\}, then $|S|=4^{4}=256$, and any two different answers in $S$ have at most 3 questions the same. Removing 6 elements from $S$, when the remaining 250 answers are each chosen by 8 people, there are a total of 2000 test papers, and among any 25 test papers, there are always 4 test papers whose answers are all different, and since they are all in $S$, they of course meet the requirements of the problem, indicating that $n=25$ can satisfy the requirements of the problem.
In summary, the minimum value of $n$ is 25.