Problem - 4413
How many different ways to write a positive integer $n$ as a sum of $m$ different positive integers? Different sequences are treated as distinct.
This is a special case of integer partition. Under these two conditions (fixed number of partitions and distinct sequences), this problem can be modeled as counting the positive integer solutions to $$x_1 + x_2+\cdots + x_m = n$$
whose answer is $\boxed{\binom{n-1}{m-1}}$.