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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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Suggested plan: figure out how many ways to choose one 3-element subset of $S$; figure out how many ways to choose 3 3-element subsets of $S$; figure out how many of those ways violate the condition forbidding the union to be all of $S$. – Gerry Myerson Nov 26 '12 at 4:25
I have problem about the case $A_i\cup A_j\cup A_t \neq S$ I don't know what should I do with it – Haruboy15 Nov 26 '12 at 5:12
You should do (or at least try to do) what I said. Count the total number, and subtract off those where the union is $S$. That will leave the ones you want, where the union is not $S$. – Gerry Myerson Nov 26 '12 at 5:28
There is a little ambiguity. Is $k$ a fixed number, or do you want the sum over all permissible $k$? – André Nicolas Nov 26 '12 at 6:19
@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. – Gerry Myerson Nov 26 '12 at 22:34

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