The 2n choose n solution isn't at all intuitive to me but thinking about it in terms of 40 steps, 20 of them rightward and 20 of them downward an then looking at all distinct permutations of these 40 steps as (40!) / ((20!)^2) is intuitive to me. Then it becomes obvious that since 20 is half of 40, k and n - k are the same number (20), which coincides with the binomial coefficient n! / k!(n - k)!. But this seems like a lucky coincidence in 2 dimensions and if you extended the problem into 3D you'd do better thinking about permutations.