Quote:

2. Given a positive integer $\displaystyle n$, define a $\displaystyle k$-partition

to be a sum of $\displaystyle k$ positive integers which sum to $\displaystyle n$.

For example, $\displaystyle n=10$.

The following are $\displaystyle 4$-partitions: $\displaystyle 10 = 4+4+2+2,\; 10 = 2+2+4+4,\;10 = 1+1+1+7$.

Notice that $\displaystyle 2+2+4+4\mbox{ and }4+4+2+2$ are considered distinct. *****

Say you are a given a specific $\displaystyle n$ and given a specific value of $\displaystyle k$.

Can you find the total number of $\displaystyle k$-partitions of this integer, with a formula?

***** This is done to simplify this problem.

When these are not considered distinct, this forms a problem from number theory

called the "partition problem", a very complicated problem.

Yes, it is. I investigated it years ago.