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**Input Description:** An integer \(n\). **Problem:** Generate (1) all, or (2) a random, or (3) the next subset of the integers \(1\) to \(n\).

**Excerpt from** The Algorithm Design Manual: A subset describes a selection of objects, where the order among them does not matter. Many of the algorithmic problems in this catalog seek the best subset of a group of things: vertex cover seeks the smallest subset of vertices to touch each edge in a graph; knapsack seeks the most profitable subset of items of bounded total size; and set packing seeks the smallest subset of subsets that together cover each item exactly once.

There are \(2^{n}\) distinct subsets of an \(n\)-element set, including the empty set as well as the set itself. This grows exponentially, but at a considerably smaller rate than the \(n!\) permutations of \(n\) items. Indeed, since \(2^{20} =\) 1,048,576, a brute-force search through all subsets of 20 elements is easily manageable, although by \(n=30\), \(2^{30} =\) 1,073,741,824, so you will certainly be pushing things.