Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets?



Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets?
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*.


