Solved by Nizar Jalal

technical solution for screwy pirates

Here is solution that works for any number of pirates and any number of pirates needed to open to chest.

Let T be the total number of pirates. Let N be the number of pirates required to open the chest.

The number of locks needed would be L = (T,N-1) = (T!)/[(N-1)! * (T-N-1)!]. The number of keys each pirate would have be K = (T-1,N-1) = (T-1)!/[(N-1)! * (T-N)!].

For this specific problem, the number of locks would be 1716 and the number of keys per pirate would be 924.

I know I haven't provided an explanation of why and how this system works. We'll leave that lengthy, involved explanation to the author.

Note: the notation (x,y) is the combination notation; I can't use the conventional combination notation in plain text. (x,y) basically asks the question "how many ways can you pick y objects from a group of x objects?" (x,y) = (x!)/[y! * (x-y)!]