What is the number of ways of distributing 4 identical balls among 6 different boxes?

In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.

The number of ways to partition $n$ distinguishable balls into $k$ indistinguishable cells such that no cell is empty is given by $S(n,k)$ where $S(n,k)$ is the Stirling number of the second kind. Thus in your case the number of ways is given by $$ S(4,1)+S(4,2)+S(4,3)+S(4,4)=1+7+6+1=15. $$ Clearly $S(4,1)=1=S(4,4)$ and $S(4,3)=\binom{4}{2}=6$ since we choose which two balls go into the same box. Finally $S(4,2)=\frac{1}{2}(2^4-2)=2^3-1=7$ since the collection of pairs $(A,B)$ where $A, B$ partition $[4]$ (so $B=A^c$) is specified by requiring that $A$ be a nontrival subset of $[4]$. Since the boxes are indistinguishable we divide by $2$.