91Ó°ÊÓ

Stirling numbers of the second kind, part 1. Let \(S(n, k)\) be the number of ways to partition a set of \(n\) elements into \(k\) nonempty subsets. A partition of a set \(A\) is a collection of subsets of \(A\) such that each element of the set \(A\) must be an element of exactly one of the subsets. The order of the subsets is irrelevant as the partition is a collection (a set of sets). For example, the partition \\{\\{1\\},\\{2,3\\},\\{4\\}\\} is a partition of \\{1,2,3,4\\} . \\{\\{4\\},\\{1\\},\\{2,3\\}\\} is the same partition of \(\\{1,2,3,4\\} .\) (a) Find \(S(10,1) .\) (b) Find \(S(3,2)\). (c) Find \(S(4,3)\). (d) Find \(S(4,2)\). (e) Find \(S(8,8)\)