An intuitive motivation for the solution in the article (2n choose n). For an n*n grid you have to you will take 2n steps, n "over" and n "down". All that matters is the order of the steps. So if you think of there being 2n "slots", you have to pick n to be "over", and the rest are forced to be "down". So it's n choose 2n indeed.

You can also think of it another way, without using the formula combinations, and only the fact that there are n! permutations of n objects. We can think of this a permutation of 2n items, made up of two groups of n identical items each. Using (2n!) will overcount, due to the fact that each of the "over" steps are identical, and similarly for the "down" group. We have cut down our answer by dividing out all of the repeated sequences. There will be n! redundancies for all the ways we can permute the "over" group and, the same for the "down" group. So this results in (2n!) / (n! * n!), which is exactly equal to 2n choose n. See [1] which explains permutations with repetion this in general. [Note: We pretty much re-derived the formula for combinations!]

[1] https://brilliant.org/wiki/permutations-with-repetition/