2. 22 Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$.
[Solution] For $k=0,1,2,3,4$ define
$F_{k}=\{A: A \subseteq S,|A|=31, A$'s element sum $\equiv k(\bmod 5)\}$, where $|A|$ denotes the number of elements in set $A$. Since $31 \equiv 1(\bmod 5)$, if $\left\{x_{1}, x_{2}, \cdots, x_{31}\right\}$ is in $F_{0}$, then $\left\{x_{1}+k, x_{2}+k, \cdots, x_{31}+k\right\}$ is in $F_{k}$ (here, if $x_{i}+k$ exceeds 1990, then use $x_{i}+k-1990$ instead). Clearly, this is a one-to-one correspondence from $F_{0}$ to $F_{k}$. Therefore,
$$
\left|F_{0}\right|=\left|F_{1}\right|=\left|F_{2}\right|=\left|F_{3}\right|=\left|F_{4}\right| \text {. }
$$

Since $F_{0}, F_{1}, F_{2}, F_{3}, F_{4}$ have no common elements,
$$
\sum_{k=0}^{4}\left|F_{k}\right|=C_{1990}^{31},
$$

thus
$$
\left|F_{0}\right|=\frac{1}{5} C_{1990}^{31} \text {. }
$$