A grasshopper starts at the origin and is equally likely to hop north,s,e,w. What is the probability that it's coordinates will be 0,0 after 4 hops? The grasshopper must hop in all $4$ directions (North, South, East, and West) to get back to the origin after $4$ hops. Therefore, I did: $\frac{(4 \cdot 3 \cdot 2 \cdot1)}{4^4} = .09375$. However, the answer key tells me that the answer should be $\frac{36}{256} = .140625$, which is exactly six times my answer. I'm not sure why my answer would be incorrect (since the grasshopper can hope either North, South, East, or West the first time, and if it hopped North, it can hop South, East, or West; if it hopped East, it can hop South or West, and so on.) it covers all the different orders that the grasshopper could hop in.

There are indeed $24$ ways to return to $(0,0)$ if we move in all four directions, but we can also get there moving only north and south or only east and west. Each possibility gives us $\binom{4}{2}=6$ options. Hence there are $24+12=36$ ways to return to $(0,0)$ in $4$ moves. On the other hand there are $4^4$ possible movement sequences.

Final answer is hence $\frac{36}{4^4}=0.140625$