#### Answer

The number of different ways to choose and rank the three best movies is 6840.

#### Work Step by Step

We need to choose 3 movies from a list of 20 movies. The order in which the movies are chosen matters because the movies should be selected according to preference. Therefore, we need to find the number of permutations of 20 things taken 3 at a time.
Use the formula,
${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
Where $ n=20,r=3$
$\begin{align}
& {}_{20}{{P}_{3}}=\frac{20!}{\left( 20-3 \right)!} \\
& =\frac{20!}{17!}
\end{align}$
Solve after to get,
$\begin{align}
& _{20}{{P}_{3}}=\frac{20\times 19\times 18\times 17!}{17!} \\
& =20\times 19\times 18 \\
& =6840
\end{align}$