Does a 10element set have 30 3element subsets such that each pair is in two of these 30 subsets?
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This asks for a $(10,3,2)$ balanced incomplete block design. These are known. There are $960$ different designs with those parameters up to isomorphism according to the CRC Handbook of Combinatorial Designs. 


If you prefer websites to books, you can go to http://designtheory.org/database/tdesigns/ and scroll down to the line that starts 2 10 30 9 3 2, and click on "download" to receive the 960 designs. 

