Four. (50 points) Let $S=\{1,2, \cdots, 2025\}$. If the sum of the elements of a 51-element subset of $S$ is divisible by 5, then it is called a "good subset" of $S$. Find the number of good subsets of $S$.


Four, divide all 51-element subsets of set $S$ into five classes $S_{0}, S_{1}, \cdots, S_{4}$, where $S_{i}(i=0,1, \cdots, 4)$ consists of 51-element subsets whose sum of elements is congruent to $i$ modulo 5.

It is easy to see that for any $i, j (0 \leqslant i, j \leqslant 4, i \neq j)$, there is a unique $m (0 \leqslant m \leqslant 4)$ such that $(i+m)-j$ is divisible by 5.
We establish the following correspondence:
For any $A \in S_{i}$, let $A=\left\{x_{1}, x_{2}, \cdots, x_{51}\right\}$, and define $A^{\prime}=\left\{x_{1}^{\prime}+m, x_{2}^{\prime}+m, \cdots, x_{51}^{\prime}+m\right\}$, where for $k=1,2, \cdots, 51$,
$$
x_{k}^{\prime}=\left\{\begin{array}{ll}
x_{k}, & x_{k}+m \leqslant 2025 ; \\
x_{k}-2025, & x_{k}+m>2025 .
\end{array}\right.
$$

Since the sum of elements in set $A$ is congruent to $i$ modulo 5, by the choice of $m$, the sum of elements in $A^{\prime}$ is congruent to $j$ modulo 5, thus $A^{\prime} \in S_{j}$. Therefore, the correspondence above gives a mapping from $S_{i} \rightarrow S_{j}$.

This mapping must be injective, i.e., for $A, B \in S_{i}, A \neq B$, we have $A^{\prime} \neq B^{\prime}$. This can be shown by arranging the elements of $A$ and $B$ in ascending order and noting that for $1 \leqslant x_{k}, y_{l} \leqslant 2025$, we have
$$
x_{k}+m-2025 \neq y_{l}+m.
$$

Thus, if $A^{\prime}=B^{\prime}$, then $A=B$.
Since the above mapping is injective, we have $\left|S_{i}\right| \leqslant\left|S_{j}\right|$. By the symmetry of $i$ and $j$, we also have $\left|S_{j}\right| \leqslant\left|S_{i}\right|$. Therefore, $\left|S_{i}\right|=\left|S_{j}\right|$.
Thus, $\left|S_{0}\right|=\left|S_{1}\right|=\left|S_{2}\right|=\left|S_{3}\right|=\left|S_{4}\right|$.
Since $S$ has $\mathrm{C}_{2025}^{51}$ 51-element subsets, we have $\left|S_{0}\right|=\frac{1}{5} \mathrm{C}_{2025}^{51}$.
(Wang Jizhong, Shandong Zhangqiu No. 4 High School, 250200)