Question 85
Let $S=\{1,2, \ldots, 2021\}$, and let $\mathcal{F}$ denote the set of functions $f: S \rightarrow S$. For a function $f \in \mathcal{F}$, let
$$
T_{f}=\left\{f^{2021}(s): s \in S\right\}
$$
where $f^{2021}(s)$ denotes $f(f(\cdots(f(s)) \cdots))$ with 2021 copies of $f$. Compute the remainder when
$$
\sum_{f \in \mathcal{F}}\left|T_{f}\right|
$$
is divided by the prime 2017, where the sum is over all functions $f$ in $\mathcal{F}$.
$$
T_{f}=\left\{f^{2021}(s): s \in S\right\}
$$
where $f^{2021}(s)$ denotes $f(f(\cdots(f(s)) \cdots))$ with 2021 copies of $f$. Compute the remainder when
$$
\sum_{f \in \mathcal{F}}\left|T_{f}\right|
$$
is divided by the prime 2017, where the sum is over all functions $f$ in $\mathcal{F}$.