Counting marbles.(partitions into n sumands) In how many ways can I divide a pile of n identical marbles into k  piles of marbles such that each pile has at least one marble? The piles are not different.  I tried to use combinations with repetition  and then do some sort of recursive definition but I find it too complicated. Is there a simpler way to make sense of it?
Regards.
 A: First you have to place a marble in each pile. Since the marbles are all the same and we aren't counting piles differently, this simply removes $k$ marbles from the original $n$. Next, we have to take the remaining $n-k$ marbles and place them into the $k$ piles in some way. This is the same as counting the ways of partitioning $n-k$ as a sum of nonnegative integers, where the number of summands of the partition is less than or equal to $k$. In particular, the number of ways to place your marbles (call it $N$) will satisfy $N\leq p(n-k)$ (here $p(m)$ is the partition function seen in the wiki link). I'm not sure if there is a known closed form for the number of partitions of a number $n - k$ into $k$ or fewer summands; perhaps someone else with more experience in that area of number theory would be able to provide a reference or formula with that information (or even just confirm or deny existence of such a formula).
A: You want the number of partitions of $n$ Into $k$ parts. There is no easy formula, though there is a recurrence.
