Zodiac Sign - 12 (Sun position can be in any of the 12 signs)
Houses - 12 (Ascendant can start in any of the 12 houses)
Planets - 10 (Planets can be in any of the 12 signs)
Ascendant or Rising Sign - 360 degrees
Moon Sign - 12
Rulership & Aspects 12 x 10 x 10 = 1200
So taking just these factors into account there are
12 x 12 x 12^10 x 360 x 12 x 1200 = 1.4 x 10^27
As we all know, we also need to consider the house systems and aspects (which are more complicated to calculate) to further narrow the number of possible birth charts.