# Number of ways to lay out a checkers board

If I have a checkers board (8x8) and 16 black pieces , 16 white ones (this isn't the usual checkers).

Assume that these pieces can be placed anywhere on the board.

How can I calculate the number of ways the board can be laid out?

(it's trivial to say that each position may be a white piece , black piece or empty, but I have an upper bound on the number of checker pieces).

edit:

Considering the upper bound (ie 16 or less black/white piece) how does the number of ways change?

Now, assume that pieces can upgrade to kings. Needless to say, when you gain a king, you lose a normal piece, which preserves the the max 16 pieces rule.

How can I calculate the number of ways the board can be laid out considering the kings?

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$${64\choose16}{48\choose16}$$ Multiply by $2^{32}$ to get the answer when there can be kings.
If you have at most 16 black and at most 16 white, you get $$\sum_{m,n=0}^{16}{64\choose m}{48\choose n}$$ which is $$\left(\sum_0^{16}{64\choose m}\right)\left(\sum_0^{16}{48\choose n}\right)$$