Split $n$ balls to $k$ boxes I have $n$ different balls $(1,2,..., n)$ and $k$ different boxes $(1,2,...,k)$. I want put all balls to boxes, but if $i$th ball has smaller number than $j$th ($i < j$), then $i$th ball must be put to box with number not bigger than box with $j$th ball. In which many ways I can do that?

Once you have decided how many balls go into each of the different bins, there is only one way to distribute the balls. For instance, let's say $n = 4$ and $k = 3$. Then if we want two balls in bin $1$, no balls in bin $2$, and two balls in bin $3$, then we must have ball $1$ and $2$ in bin $1$ and ball $3$ and $4$ in bin $3$. That means that this is a stars and bars problem.
The answer is $\binom{n+k}{n}$.
