N containers and M balls, each container has it's own capacity let's say (c1, c2, c3, ....cN), we well distribute the balls between containers where the whole capacity of containers is bigger than the number of balls, in how many ways we can do that???
Asked
Active
Viewed 37 times
0
-
It seems difficult to find a concise general solution. The best I can think of is an inclusion-exclusion over all ways the distribution can exceed the capacity of some containers. – Kevin Long May 17 '18 at 16:18
-
can you give me more details about inclusion-exclusion solution – phonex May 17 '18 at 20:41
-
For $S\subseteq [N]$, let $a_S$ denote the number of ways to distribute the balls such that if $i\in S$, then container $i$ is over capacity. Then by inclusion-exclusion, you get $\sum_{S\subseteq [N]} (-1)^{|S|}a_S$. – Kevin Long May 17 '18 at 23:37