Greetings,

I have a problem similar to the following:

Suppose at some call center, there are a lot of people available to take calls. They are grouped according to their average success rate. There are groups of call takers with people in each respectively. Each group has a different average success rate. If each person takes a call at exactly the same time, what is the probability that they will succeed with exactly callers (across the entire call center)?

My approach is to sum over all possible solutions to a diophantine equation in variables where the sum must equal . Hence I have:

Any ideas for another approach that might be solvable in polynomial time? Or possibly a book I might take a look at that might offer some clue?

Edit: I am not a statistician. I have some experience with combinatorics, and I would prefer a combinatorial approach, but in this instance, it seems like statistics may be a more appropriate field to estimate the results. I really don't know enough about statistics to know where to begin looking, though.