Answer
The number of different ways in which the director can cast the male role is 336.
Work Step by Step
The director will cast 3 male roles from a group of 8 actors. The order in which the actors are chosen matters because Tony, Riff, and Bernardo each have different roles. Thus, we need to find the number of permutations of 8 things taken 3 at a time.
Apply the formula,
${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
Where $ n=8,r=3$
$\begin{align}
& {}_{8}{{P}_{3}}=\frac{8!}{\left( 8-3 \right)!} \\
& =\frac{8!}{5!}
\end{align}$
Solve further to get:
$\begin{align}
& _{8}{{P}_{3}}=\frac{8\times 7\times 6\times 5!}{5!} \\
& =8\times 7\times 6 \\
& =336
\end{align}$
