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Chapter 11: Problem 46

In a production of West Side Story, eight actors are considered for the male roles of Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?

Short Answer

Expert verified
There are 336 different ways to assign the actors to the three roles.

Step by step solution

01

Understand Permutations

Permutations refer to the number of ways a set of objects can be arranged. When dealing with permutations, the order of arrangement matters. In this case, each role is a unique arrangement.
02

Calculate Permutations

We have 8 actors and 3 roles to fill. We are interested in how many ways we can assign these roles, so we need to calculate the permutation of selecting 3 actors from 8. This is represented mathematically as \(P(n,r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of options, \(r\) is the number of selections to be made, and \(!\) denotes factorial.
03

Plug In Numbers

We fill in the given values into the permutation formula: \(P(8,3) = \frac{8!}{(8-3)!}\). Simplifying the subtraction gives: \(P(8,3) = \frac{8!}{5!}\).
04

Compute Factorial

Next, compute the factorial: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\) and \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). So, \(P(8,3) = \frac{40320}{120}\).
05

Simplify

Finally, simplify the fraction to get the number of permutation allocations: \(P(8,3) = 336\).

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