## In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

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Below is a table of values of numbers of admissible subsets given X = n: n Number of admissible subsets References [1] E. Bolker, The finite Radon transform, Contemporary Mathematics 63 (1987) [2] R. Abstract Single-track Gray codes are cyclic Gray codes with codewords of length n, such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track. Olympiad Combinatorics Brualdi (Introductory Combinatorics, 3rd. ed. 1999, p. 51) and Chen & Koh (Principles and Techniques in Combinatorics, 1992, pp. 34-37) explicitly allow infinite repetition. Suppose that the marriage condition fails, i.e., that for some subcollection W 0 {\displaystyle W_{0}} of S {\displaystyle S} , | W 0 | > | ⋃ A ∈ W 0 A | . {\displaystyle \textstyle |W_{0}|>|\bigcup _{A\in W_{0}}A|.} Suppose, by way of…

## Brualdi, 2004. R.A. BrualdiIntroductory combinatorics. (Fourth Ed), Prentice Hall, New Jersey (2004). Google Scholar. Cuoco, 2006: Cuoco, Al. (2006). Preparing

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