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Combinatorics

Let \(S= \{1,2,.....,n\}\) and let \(T\) be the set of all ordered triples of subsets \(S\), say \((A_{1}, A_{2}, A_{3})\), such that \( A_{1} \cup A_{2} \cup A_{3} = S\). Find the cardinality of \(T\) in terms of \(n\). (For example, if \(S = \{1,2,3\}\) and \(A_{1} = \{1, 2\}, A_{2} = \{2, 3\}, A_{3} = \{3\}\) then one of the elements of T is (\(\{1, 2\}, \{2, 3\}, \{3\})\). )

Note by Dhrubajyoti Ghosh
2 weeks, 1 day ago

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