For the median of the $5$ books to be $7$ we need that $3$, $4$ or $5$ of them choose book $7$ as their favorite.

There are $\displaystyle \binom{5}{3} \times 6^2$ was to have $3$ of them choose book $7$ as their favorite, since we need to choose which $3$ of the $5$ chose it, and we also need to choose some of the other $6$ books for each of the other $2$ people.

Similarly, there are $\displaystyle \binom{5}{4} \times 6$ and $\displaystyle \binom{5}{5} \times 6^0$ ways for $4$ of them or $5$ of them, respectively, to choose book $7$ as their favorite. 

Thus, there are $\displaystyle \binom{5}{3} \times 6^2 + \binom{5}{4} \times 6^1 + \binom{5}{5} \times 6^0$ ways to have the median of the books to be $7$.