This question was inspired by Rush Hour game:
You have a 6x6 grid, 12 pieces of size 2, and 4 pieces of size 3. A piece can be placed on the grid either horizontally or vertically. The pieces can't overlap. Note that when all the pieces are placed on the grid they take up all the space.
Considering that pieces of the same size are identical, how many different ways are there to place all the pieces on the grid?
What if the placements are considered identical up to a rotation/reflection?
Is there a solution that can be easily generalized to grids and pieces of any size? If not, is there a good estimate (upper and lower bounds) for the general case?