Problem 46 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) 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?

Step-by-Step Solution

Answer

The director can cast the male roles in 336 different ways.

Step 1: Identification

We first identify the parameters for our permutation formula. We have n = 8 (The total actors) and r = 3 (The total roles).

Step 2: Apply the permutation formula

Plug the values into the permutation formula: \( _nP_r = \frac{n!}{(n - r)!} \) => \( _8P_3 = \frac{8!}{(8 - 3)!} \).

Step 3: Simplify the factorial

We simplify the numbers as the factorials. Factorial means to multiply a series of descending natural numbers. 8! = 8*7*6*5*4*3*2*1 and 5! = 5*4*3*2*1. After you find the value for 8! and 5!, divide 8! by 5!.

Step 4: Solve

The calculation leads to (40320/120 = 336). There are 336 ways the director can cast the roles of the play under the given conditions.