Originally Posted by

**Jameson** I am studying for the Putnam competition in December and I want just a hint at how to do this problem, not a full solution please. I'm trying to develop a good methodology to approaching proofs.

A1. Determine, with proof, the number of ordered triples $\displaystyle (A_1,A_2,A_3)$ of sets which have the property that

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

(ii)$\displaystyle A_1 \cap A_2 \cap A_3 \ne \emptyset$. Express the answer in the form of $\displaystyle 2^a3^b5^c7^d$, where a,b,c, and d are non-negative integers.

What's the best way to approach this?