Find a way to go home

Two possible way can be just like these picture:

One possible way is $(\rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \uparrow \uparrow \uparrow \uparrow )$

Another possible way is $(\rightarrow \rightarrow \uparrow \uparrow \rightarrow \rightarrow \uparrow \uparrow \rightarrow \rightarrow )$

In every way we get $10$ arrow where $6$ are right arrow $(\rightarrow)$ and $4$ are up arrow $(\uparrow)$ .

Finally we can arrange $10$ arrow where $6$ is same $(\rightarrow)$ another $4$ is same $(\uparrow)$ .

So total way is $=\frac{10!}{6!\cdot 4!}=\boxed{210}$

---

Another solution is:

Total way $=^{10}C_{6}=^{10}C_{4}=\boxed{210}$