How many different (7,7,3,3,1) BIBD are there on vertex set[7]? Note that two such designs are different if their sets of blocks are different. That is, we do not require that the BIBDs be non-isomorphic; we simply require that they be non-identical.
I have found that we use construct the fano plane shape to illustrate this problem but counting part is difficult. I found the two ways to construct fano plane but any suggestion on how to count them up?
Let the vertices be 1, 2, 3, 4, 5, 6, 7. The blocks containing 1 must be 1ab, 1cd, and 1ef, where $\{{\,a,b,c,d,e,f\,\}}=\{{\,2,3,4,5,6,7\,\}}$. So the first question is, how many ways can you partition a 6-set into 3 2-sets?
Now, the block containing a and c must contain either e or f. Once you have decided which, the rest is forced, e.g., if ace is a block, the other blocks must be adf, bcf, and bde.
That should give you all you need.