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Even if you did have all the pieces, it would be a very complicated problem because how you attach one piece will affect how many different ways you can connect the next piece.

If you are just stacking them (second one on top of first, etc) then it's a standard permutation and the answer is 20! which is 2,432,902,008,176,640,000 so billions doesn't even scratch the surface.

2^20 is just how many different combinations of pieces there are. As in, there are 2^20 different ways you could pick a subset of the pieces to attach. (2^20 - 1 is if you skip the case where you pick none of them).

20! may be more what you were remembering, but that doesn't take into account the different shapes and how they can connect differently.