Putnam 1985, A1

Determine, with proof, the number of ordered triples \( (A_{1}, A_{2}, A_{3}) \) of sets which have the property that

(i) \( A_{1} ∪ A_{2} ∪ A_{3} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), and
(ii) \(A_{1} ∩ A_{2} ∩ A_{3} = ∅\),

where \(∅\) denotes the empty set.

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