Let there be an urn containing $N$ distinct types of balls. Each type of ball has a different number printed on it e.g. balls of type $n$ have the real number $n$ printed on them. There are $K_n$ balls of type $n$ in the urn.
Balls are drawn without replacement from the urn. Let $C \in \mathbb{R}$, and let $S_{q}$ be the sum of the numbers printed on the balls that have been drawn from the urn after $q$ balls have been drawn. Furthermore, all balls that if drawn on the $(q+1)$th draw would result in $S_{q+1}>C$ are removed from the urn. This is true for all $q=0,1,2,...$. Thus the sum of the numbers on the balls that have been drawn never exceeds $C$.
What is the probability of drawing $k_n$ balls of type $n$ from the urn following this process?
Let's say I multiply $K_n$ by a positive integer $\alpha$ for all $n$, and also I multiply $C$ by $\alpha$. Let $P(\alpha k_n)$ be the probability of drawing $\alpha k_n$ balls of type $n$ from this larger urn. What is $\lim_{\alpha \rightarrow \infty} P(\alpha k_n)$?