Chapter 10 - Review Exercises - Page 1126: 81

The male actors can be selected in 116,280 ways for four roles.

Work Step by Step

We know that the order in which the four actors are selected makes a difference as the all the actors would be cast for a different role. Since the order matters, we use permutations. The four officers are to be selected from a club of twenty members. So, $n=20,r=4$. Hence, \begin{align} & _{20}{{P}_{4}}=\frac{20!}{\left( 20-4 \right)!} \\ & =\frac{20!}{16!} \\ & =\frac{20\times 19\times 18\times 17\times 16!}{16!} \\ & =116,280 \end{align}

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