TITLE: Split a number into parts
QUESTION [5 upvotes]: In how many ways can a natural number $n$ be split into $m$ natural numbers (parts) where each part is less than $n$, the parts don't necessarily have to be equal, and all of them add up to $n$?

REPLY [3 votes]: When summands are allowed to be zero, these are known as weak compositions of integer $n$ having $m$ parts.
They are readily counted by the "stars-and-bars" technique, as Rus May indicated.  This is a frequently-asked-question here.
The expressions where we are required to discount summations that differ only in regard to the order of otherwise the same summands are called partitions of integer $n$ having at most $m$ parts.  For these there are no simple binomial coefficient counting formulas, although work by Ramanujan and others led eventually to a "closed form" of sorts.  Therefore the counting of integer partitions is more difficult than of integer compositions, despite the overall similarity of problems.