Assuming a $3\times3$ area is given. Furthermore there are sub areas given, e.g.:
| O | O | O |
|---|---|---|
| O | O | O |
| O | O | O |
| X | |
|---|---|
| X | X |
| X | X |
|---|---|
| X |
| X | X | X |
|---|
The aim is to arrange these sub areas so that they are not overlapping and the entire $3\times3$ area is used. How many combinations are possible? Rotating is of course allowed.
For the given example it is quite easy to solve by trying out. But I am looking for a solution which can be used for any $M\times N$ area and sub areas. (It can be assumed that the sum of the sub areas is equal to the big area)
I am even lacking of an idea how to model this problem mathematically. Any idea or hint how to solve such class of problems?
Tiling problems like this are notoriously difficult, evading any sort of general symbolic approach or in most cases even algorithms with efficient runtimes. You are not going to find any sort of general formula or rule that will solve such problems, only specific techiques useful in certain situations.
In practice, I would convert these kinds of problems into one which can be tackled by an off-the-shelf SAT solver (one variable for each position of a tile, clauses enforcing that every tile occupies only one position and that no two overlapping positions occur simultaneously); this method often has good success with resolving computationally-intensive tiling problems, despite forgetting much of the geometric structure associated to the problem.