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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).


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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up vote 2 down vote accepted

$${64\choose16}{48\choose16}$$ 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 $$\left(\sum_0^{16}{64\choose m}\right)\left(\sum_0^{16}{48\choose n}\right)$$

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Thanks. I get the first part. But I don't see how * 2^32 solves the kings problem? What about the upper bound problem? – Mazyod Sep 17 '12 at 5:32
Given the first part, on each of the 32 occupied squares, you can choose to replace the piece with a king, or not. I'll edit in something about the "upper bound" problem, unless someone beats me to it. – Gerry Myerson Sep 17 '12 at 5:35

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