Final answer:
There are 116,280 different ways for a director to cast the roles of Mark, Roger, Angel, and Collins from 20 male actors for the musical Rent. This is calculated using permutations since the roles are distinct and order matters.
Step-by-step explanation:
The number of different ways a director can select from 20 male actors to cast the roles of Mark, Roger, Angel, and Collins in the musical Rent is 116,280.
To calculate this, we use the concept of permutations since the order in which the actors are selected matters, as each role is unique. With 20 actors available for the first role, 19 for the second, 18 for the third, and 17 for the fourth, we calculate the permutations by multiplying these numbers: 20 × 19 × 18 × 17 = 116,280 different permutations.
This calculation illustrates the process of casting in theater, and how a director must consider the vast possibilities when selecting actors for specific roles. It underscores the complexity and importance of casting decisions in producing a successful show like Rent, which profoundly impacted theater with its representation of diverse characters and subject matter.