| Distribution of | Restriction | ||||
| $k\text{ Balls}$ | $n\text{ Boxes}$ | None | $\le 1$ | $\ge 1$ | $=1$ |
| $distinct$ | $distinct$ | $$n^k$$ | $$(n)_k$$ | $$n!S(k,n)$$ | $$n!\text{ or }0$$ |
| $identical$ | $distinct$ | $${n+k-1}\choose k$$ | $$n\choose k$$ | $${k-1}\choose{n-1}$$ | $$1\text{ or }0$$ |
| $distinct$ | $identical$ | $$\sum_{i=1}^{n}S(k,i)$$ | $$1\text{ or }0$$ | $$S(k,n)$$ | $$1\text{ or }0$$ |
| $identical$ | $identical$ | $$\sum_{i=1}^{n}P(k,i)$$ | $$1\text{ or }0$$ | $$P(k,n)$$ | $$1\text{ or }0$$ |
$(n)_k=n(n-1)(n-2)…(n-k+1)=k!{n\choose k}$
$S(k, n)$ is a Stirling number of the second kind:
$$S(k,n)=\dfrac{1}{n!}\sum_{i=0}^{n}(-1)^{i}{{n}\choose{i}}(n-i)^k$$
$P(k, n)$ is the number of partitions of $k$ into $n$ parts