This problem sounds like it'll take a while. I think I'll start by finding an upper bound (ie
something guaranteed to be an overestimate.) I'll round up the numbers and give the
estimate in brackets.
Suppose we have 32 pieces on the chessboard. Then there are 64 choose 32 ways of covering
squares with them (2*10^18) . Two of these are the kings, so we have 32x31 (700) ways of
putting these on the covered squares. Then we have 8 pawns for each player, so we have 30
choose 8 for White and 22 choose 8 for Black (6*10^7, 320000). But these pawns could each
be in one of 6 states (promoted to one of 4 things, unpromoted, captured), so that gives a
further 6^16 (3*10^12) possibilities. Now among the remaining 14 pieces, we have three
kinds of piece occurring twice, so we get 14!/8 (1.1*10^10) ways of arranging them
(eliminating ones where two alike pieces are swapped around.) But any of these could have
been captured, so we have to multiply by 2^14 (17000).
Multiplying all these rounded up numbers together, you get around 10^57. So at least we
know the real answer isn't THAT big.