I'm writing a play that features a lot of randomization, which will mean that it is different every time, and I'm trying to calculate the number of performance possibilities.
- It features a number of scenes, which will be performed in a different order each performance.
- The roles within each scene are to be played by different actors each time.
- The cast size is also variable!
So there are X amount of scenes, Y amount of roles within the scenes, and Z amount of actors to be filled in each role. (For example: 38 scenes, with 77 total speaking roles to be divided among 9 actors.) The speaking roles would ideally be evenly distributed among the actors.
How would I calculate the number of performance possibilities this play can hold?
Hopefully I'm making myself clear! Thanks for any help you can offer!
With this data
there are $38! = 5.23 \times 10^{44}$ orders for the scenes, even before assigning actors. That's more than $5$ followed by 44 $0$'s.
For convenience estimating, I'll work with $72$ rather than $77$ roles, since that's a multiple of $9$ - each actor will get $8$ roles. There are $72!/(8!)^9 \approx 2.2 \times 10^{62}$ ways to assign the roles.
When you multiply those together you get a number with more than $109$ digits. A googol has a mere $100$ zeroes. This is about a billion googol.