In n-space the problem is almost as easy as in 2-space if you can only go in forward in each dimension. If you can go in both directions and you want to find th

In n-space the problem is almost as easy as in 2-space if you can only go in forward in each dimension. If you can go in both directions and you want to find the number of non self intersecting paths, the problem is hard even in 2-space (as far as I can tell). If you allow paths to be self-intersecting, then the count is obviously infinite. Note that the actual interview question asks you to list all paths, not count them. To count paths in 2-space from (0,0) to (5,6), you have the operations "X++" and "Y++" which go right and up, respectively. Each increasing path from (0,0) to (5,6) has to be some permutation of 5 times X++ and 6 times Y++. So the count is (5+6)!/(5!6!) where the exclamation mark denotes factorial. This extends to higher dimensions by adding an additional operation "Z++". Then the count to go from (0,0,0) to (5,6,7) is (5+6+7)!/(5!6!7!).