For some math fun! It seems to me like the extreme upper end of possible positions would be: (64 choose 5) * (10^5) * (59 choose 2) * (2!). Logic:
(64 choose 7) - The number of distinct 7 square selections on the board. This is where our pieces go.
(10^5) - For each piece there are a maximum number of 10 possible choices (queen/rook/pawn/knight/bishop of black or white)
(59 choose 2) - There are 59 remaining squares, we need to pick 2 for the kings to be on.
(2!) - The combinations pieces of the two squares for the kings - obviously just two.
This is mathematically equivalent to (64 choose 7) * (10^5) * (7 choose 2) * (2!) for those that might find that method of counting more intuitive.
This seems to give an extreme upper bound of ~2,609,000,000,000,000 position. Our real number (423,836,835,667,331) is 16.2% of that. That means 83.8% of all possible positions are invalid. Seems just about right.
(64 choose 7) - The number of distinct 7 square selections on the board. This is where our pieces go.
(10^5) - For each piece there are a maximum number of 10 possible choices (queen/rook/pawn/knight/bishop of black or white)
(59 choose 2) - There are 59 remaining squares, we need to pick 2 for the kings to be on.
(2!) - The combinations pieces of the two squares for the kings - obviously just two.
This is mathematically equivalent to (64 choose 7) * (10^5) * (7 choose 2) * (2!) for those that might find that method of counting more intuitive.
This seems to give an extreme upper bound of ~2,609,000,000,000,000 position. Our real number (423,836,835,667,331) is 16.2% of that. That means 83.8% of all possible positions are invalid. Seems just about right.