Not a homework question, promise. I'm studying for the LSATs (the Law School Admissions Test), which has a section of logic games that are basically set-covering / mapping problems. They're not designed to be solved completely, and instead rely on you making quick deductions to limit the number of possibilities to answer a multiple choice question about a particular contingency.

Very frequently, a problem will be set up along the lines of: "Exactly six piano classes are given sequentially, in which nine students are taught," and you'll have to sort out the possible sequences of students, keeping in mind that some students may be taught simultaneously. This therefore gives rise to a number of possibilities, among them:

- 1 class contains 4 students and 5 classes contain 1 student. (4-1-1-1-1-1)
- 1 class contains 3 students, 1 class contains 2 students, and the rest contain 1 student (3-2-1-1-1-1), etc.

It would be nice to be able to calculate exactly how many options exist without having to count them by hand. It seems like the best way to formulate the problem is:

*How many unique ways are there to express a whole number *n* as the sum of *m * nonzero whole numbers?*

I'm not a math student, and I could be tagging this incorrectly. I'd appreciate any help in solving the problem, or pointers to similar problems that may have already been posted.

Thanks, Aaron

nameof the problem category is a good first start! $\endgroup$ – awenner Sep 19 '10 at 23:46