Recall that the Shannon entropy of a random variables $X$ taking values in a finite set $S$ is given by
\[H[X]=-\sum_{x \in S} \operatorname{Pr}[X=x] \log _{2} \operatorname{Pr}[X=x] .\]
(We set $0 \log _{2} 0=0$.) For a pair of random variables $(X, Y)$ taking values in the finite set $S \times T$, we write
\[
\begin{array}{l}
H[X \mid Y=y]=-\sum_{x \in S} \operatorname{Pr}[X=x \mid Y=y] \log _{2} \operatorname{Pr}[X=x \mid Y=y] \\
\text { and } H[X \mid Y]=-\sum_{y \in T} \operatorname{Pr}[Y=y] H[X \mid Y=y] .
\end{array}
\]
Now, consider an $1024 \times 1024$ chess board. Suppose $1024$ rooks are placed one after another randomly at distinct locations on a $1024 \times 1024$ chess board so that no rook attacks another: that is, the $i$-th rook $(i=1,2, \ldots, 1024)$ is placed at a location chosen uniformly from among the available possibilities so that it does not attack any of the previously placed rooks. Let $R_{i}$ be the row number of the $i$-th rook and $C_{i}$ its column number. What is $H\left[R_{513}, C_{513} \mid R_{1}, R_{2}, \ldots, R_{512}\right]?$
- $\log _{2} 513$
- $9$
- $10$
- $19$
- $81$