Problem 46 - Algebra and Trigonometry

Question

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

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

Step 1: 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.

Step 2: 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.

Step 3: Plug In Numbers

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

Step 4: 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, \(P(8,3) = \frac{40320}{120}\).

Step 5: Simplify

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