Combinations of Checkers?

Question

How many possible combinations of checkers boards are there? 0 to 12 black pieces, 0 to 12 white pieces. Any piece an be kinged. 32 squares.

I'll give 10 points to whomever can give me the lowest number and show how you did it.

Answer 1

There are a lot of combinations. If you assume each square is distinct, and don't allow for symmetries (i.e. 1 white piece in a corner is the same no matter what corner it's in), the answer I got was:

2308487701529742077825

Basically, my method was to first choose where to put x black pieces in 32 choose x ways. Then choose where to put y white pieces in 32-x choose y ways. Then for each piece, you need to choose whether or not to king it. This gives another 2^(x+y) ways. Then just sum over x&y from 0 to 12. Here's the formula (nCr(x,y) is x choose y, and sum(P(x),x,0,12) is the sum over P(x) as x goes from 0 to 12.):

Sum(Sum(nCr(32, x)*nCr(32-x, y)*2^(x+y), y, 0, 12), x, 0, 12)

I plugged this into my trusty TI-89, and came up with the answer:

2,308,487,701,529,742,077,825 combinations.