An Ant walk with Alvin Mak answer

An ant walks 8 steps on a number plane, starting at the origin and finishing there. How many different paths are available?



To start at (0,0) and finish at (0,0), the ant must have the same number of "north" steps as "south" steps, likewise with the "east" and "west" steps. Let N, S, E, W represent these directional steps, the problem reduces to how many ways to arrange a chain of 8 such letters such that there's equal number of N's and S's, and equal number of E's and W's, possibilities are:

1) 4N's and 4S's: 8!/(4!4!) = 70
2) 3N's, 3S's, 1E and 1W: 8!/(3!3!) = 1120
3) 2N's, 2S's, 2E's and 2W's: 8!/(2!2!2!2!) = 2520
4) 1N, 1S, 3E's and 3W's: 8!/(3!3!) =1120
5) 4E's and 4W's: 8!/(4!4!) = 70

Totalling to 4900