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| Distribution of<\/b><\/td>\n | Restriction<\/b><\/td>\n<\/tr>\n| $k\\text{ Balls}$<\/td>\n | $n\\text{ Boxes}$<\/td>\n | None<\/td>\n | $\\le 1$<\/td>\n | $\\ge 1$<\/td>\n | $=1$<\/td>\n<\/tr>\n | $distinct$<\/td>\n | $distinct$<\/td>\n | $$n^k$$<\/td>\n | $$(n)_k$$<\/td>\n | $$n!S(k,n)$$<\/td>\n | $$n!\\text{ or }0$$<\/td>\n<\/tr>\n | $identical$<\/td>\n | $distinct$<\/td>\n | $${n+k-1}\\choose k$$<\/td>\n | $$n\\choose k$$<\/td>\n | $${k-1}\\choose{n-1}$$<\/td>\n | $$1\\text{ or }0$$<\/td>\n<\/tr>\n | $distinct$<\/td>\n | $identical$<\/td>\n | $$\\sum_{i=1}^{n}S(k,i)$$<\/td>\n | $$1\\text{ or }0$$<\/td>\n | $$S(k,n)$$<\/td>\n | $$1\\text{ or }0$$<\/td>\n<\/tr>\n | $identical$<\/td>\n | $identical$<\/td>\n | $$\\sum_{i=1}^{n}P(k,i)$$<\/td>\n | $$1\\text{ or }0$$<\/td>\n | $$P(k,n)$$<\/td>\n | $$1\\text{ or }0$$<\/td>\n<\/tr>\n<\/table>\n<\/p>\n\n\n\n | $(n)_k=n(n-1)(n-2)…(n-k+1)=k!{n\\choose k}$<\/p>\n\n\n\n $S(k, n)$ is a Stirling number of the second kind<\/a>:<\/p>\n\n\n\n $$S(k,n)=\\dfrac{1}{n!}\\sum_{i=0}^{n}(-1)^{i}{{n}\\choose{i}}(n-i)^k$$<\/p>\n\n\n\n <\/p>\n\n\n\n $P(k, n)$ is the number of partitions of $k$ into $n$ parts<\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading |