How many possible starting scenarios

[yodamus]

In the advance game. I am curious as to how many starting scenarios there could be. My early guess was over 4 million. First 24 objective cards. 5 random get used. Is 42,000 plus combos. Then the 4 boards can be place in at least 100 different ways. That makes for 4 million plus.... Any one have idea the exact number ? And that is not including the fact that most of the cards don't get used either. Top notch game for never playing the same game twice.

[chaotic_iak]

Notation:

• n^r = n * n * n * ... * n, where there are r copies of n. This is exponentiation.
• n! = n * (n-1) * (n-2) * ... * 1 is the factorial.
• C(n,r) = n! / (r! * (n-r)!) is the binomial coefficient; it tells how many ways to choose r things out of a collection of n items.

There are 4! ways to arrange the boards into a 2x2 shape, and 8 ways to orient each board (4 rotations x 2 reflections), for a total of 4! x 8^4 possible forests. However, the forest has 4 rotational symmetries; rotating the whole forest around doesn't actually change what the forest is. So there are:
4! x 8^4 / 4 = 24,576
possible starting forests.

There are:
C(24,5) = 42,504
ways to choose the ecosystem cards in play.

So there are a total of:
24,576 x 42,504 = 1,044,578,304
possible starting setups for the advanced game.

For the basic game, there are 24,576 possible starting setups because the ecosystem cards are not used. For the first game, obviously there is only 1 starting setup.