According to this thread: http://www.bkgm.com/rgb/rgb.cgi?view+371 there are 18,528,584,051,601,162,496.
Also according to the GNUBG manual, the number of positions fits within a 64 bit integer.
There's the added complication of whether you count mirror positions as distinct positions or not.
Now the first calculation says:
C(24,m) X D(m+2,15-m) X D(26-m,15)
Now assuming that neither side has born off all pieces, then there is at least one piece from black which occupies a point or the bar. Assume m such points occupied where m>0. So C(25,m) ways of choosing these points
and allocating m checkers. Then there are D(m+1,15-m) ways of distributing the remaining checkers. For the other side, there are D(26-m,15) ways of distributing the remaining checkers.
This gives about 1.458e+19 possibilities, so fitting in 64 bits.
Now this is an upper bound, because there could be illegal positions included in the above (such as primes on both home boards and both sides with pieces on the bar).
Does this look right?