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Let $S=\left \{ 1;2;3;4;5;6;7;8;9 \right \}$. Ask how many way we can choose some subsets$ \left \{ A_1;A_2;...A_k \right \}$ of S, each $A_i$ has 3 element and any $A_i\cup A_j\cup A_t \neq S$

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    @Andre, I completely missed that $k$. I was, in effect, assuming $k=3$. This will require some clarification from OP, and some more thought from the rest of us.2012-11-26

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