That's a nice way to put it.

FWIW the generalization of binomial coefficient which allows you to express an n-dimensional solution is called a multinomial coefficient [0]. So in a 3d 20x20x20 box we would have (60 multichoose 20, 20, 20) paths.

Also, the wiki article doesn't mention this but the growth rate of (n multichoose k1, k2, ..., km) as we increase n but fix the ratios p1 = k1 / n, ..., pm = km / n is precisely the Shannon entropy of the categorical distribution with probabilities p1, ..., pm . The wiki article for entropy [1] states the result for the binomial coefficient, which can be written as (n choose k) = (n multichoose k, (n - k)) .

Actually a lot of basic information-theoretic results about entropy and related quantities (e.g. the properties of the Boltzmann distribution/softmax function) can be derived from similar discrete counting problems after taking a large-n limit. I don't have links at the ready but I might edit this comment if I remember places which explain this stuff.

[0] https://en.wikipedia.org/wiki/Multinomial_theorem#Number_of_...

[1] https://en.wikipedia.org/wiki/Entropy_(information_theory)#A...