AI search - build a rectangle from 12 tetris shapes how many states are possible?

You have 12 shapes:

which you can make each out of five identical squares.

You need to combine the 12 pieces to one rectangle. You can form four different rectangles: 2339 solutions (6x10), 2 solutions (3x20), 368 solutions (4x15), 1010 solutions (5x12).

I need to build the 3X20 rectangle:

My question what is the maximum number of states (i.e., the branching factor) that is possible?

My half way calculation:

The way I see it, there are 4 operations on each shape: turn 90/180/270 degrees and mirroring (turning it upside down). Then, you have to put the shape on the board, somewhere on the 3X20 board. Illegal states will be one that the shape doesn't fit in the board, but they are still states.

For the first move, you can chose each shape in 4 ways which is 4X12 ways, and then you need to multiply in the number of positions the shape can be in, and that is the number of states you have. But how can I calculate the number of positions?

Please help me with this calculation it is very important, it is not some kind of homework which I'm trying to avoid.

I think there is no easy & 'intelligent' way to list solutions (or states) to pentomino puzzles. You have to try all possibilities. Recursive programming or backtracking is the way to do it. You should check this solution that also has java source code available. Hopefully that points you to the right direction.

There is also a python solution that is perhaps more readable.