Here’s a fun little exercise where I demonstrate how quickly absurd calculating possibilities becomes when detached from reality. The topic is based on one huge assumption, and many minor ones, namely that the entire universe can be described with a single, mathematical, wave-function.
Assuming roughly 1080 ‘atoms’ in the observable universe and the location of any given one is limited by the Planck length, i.e. assuming discrete resolution instead of an infinite continuum, this gives 10180 possible locations for each.
A back of the envelope calculation then shows ( 1080 ^ 10180 ) – x possible configurations for the observable universe. where x is the number of impossible states such as all ‘atoms’ being in the same location, etc.
i.e. there exists a stupendously large number of possible states
Of course the observable universe is defined according to the location of Earth so in fact there are more ‘atoms’ than 1080 inaccessible to the Earth observer but accessible to an observer at the ‘edge’ of the Earth centered observable universe. It is assumed that physical laws and causality behaves identically at the ‘edges’.
Additionally, neutrinos, free electrons, and so on are not included in the ‘atom’ count so the real number of possible states is likely far greater.
So ( 1080 ^ 10180 ) – x is the lower bound of the number of possible states of the universe, with the actual amount likely far greater.
To give a sense of how large this is, if the relative quantities were normalized all human words ever produced would, always and forever, round down to zero in a comparison. Simply because even a perfectly ideal computer which spends all the energy in the universe could not differentiate such a tiny amount from zero. It would be impossible to even store the number of decimal places required to represent how small the quantity would be in a memory bank the size of the universe.
And depending on how you interpret the existence of such a universal wave-function, there may be vastly greater possibilities still.
Note: I put ‘atoms’ in quotes because at these scales assuming atoms as discrete, invariable, things is not quite correct physically or mathematically. A specialized audience would need a more rigorous formulation.