1985 Putnam A1

Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$
of sets which have the property that

(i) $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, 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.

通过证明确定有序三元组的数量 $(A_1, A_2, A_3)$

具有以下性质的集合

(i) $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$，并且

(ii) $A_1 \cap A_2 \cap A_3 = \emptyset$。

以 $2^a 3^b 5^c 7^d$ 的形式表达你的答案，其中 $a,b,c,d$
是非负整数。