TDSM 5.17

If the ants can move without any restriction <math>\Rightarrow<\math> There are <math>2^{2n}<\math> posibilities it can move. The ant returns to its starting points after <math>2n<\math> steps <math>\Rightarrow<\math> It has to move <\math>n<\math> steps up and <math>n<\math> steps down <math>\Rightarrow<\math> There are <math>C^{2n}_{n}<\math> posibilities it can move.

<math>\Rightarrow<\math> The posibility that the ant returns to its starting points after <math>2n<\math> steps is <\math>\frac{C^{2n}_{n}}{4^n}<\math>