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 ($\{1, 2\}, \{2, 3\}, \{3\})$. )

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$