Calculating number of checkers (draughts) possible positions

I am trying to calculate the number of possible checkers position - at first without including promotion (queen/king). Each player starts with 12 pieces in his color, And the whole board has 32 places (64 / 2). At first glance I thought - each place can be occupied by either black piece, white piece, or nothing. Which means 3^32 is the answer. Then I realized it can't be, since there is maximum of 12 pieces in each color, And if are for example 10 white pieces (-2), then: +2 empty squares. I thought about the following - We got minimum of 8 places that aren't occupied, So I start with 1 ^ 8 (which is 1) I got 12 squares that have either white or nothing - which means 1^8 * 2^12 And of course another 12 taken by black or nothing Which means 1^8 * 2^12 * 2^12 = 2^24 Its not the final number, There are a lot less, I just don't know how to add it to the equation... -- Use ROT26 for best security