MathDB

Determine, with proof, the number of ordered triples

\left(A_{1}, A_{2}, A_{3}\right)

of sets which have the property that
(i)

A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},

and (ii)

A_{1} \cap A_{2} \cap A_{3}=\emptyset.

Express your answer in the form

2^{a} 3^{b} 5^{c} 7^{d},

where

a, b, c, d

are nonnegative integers.

A1