Calculating number of checkers (draughts) possible positions

--001485f5b1dadf21f6046f525692
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 7bit

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

--001485f5b1dadf21f6046f525692--