A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
$\textbf{(A) } \frac{1}{2187}\qquad \textbf{(B) } \frac{1}{729}\qquad \textbf{(C) } \frac{2}{243}\qquad \textbf{(D) } \frac{1}{81} \qquad \textbf{(E) }  \frac{5}{243}$
Call this cube $ABCDEFGH$, with $A$ being the starting point.
Because in $7$ moves the bug has to visit the other vertices, the bug cannot revisit any vertex.
Therefore, starting at $A$, the bug has a $\frac{3}{3}$ chance of finding a good path to the next vertex, and call it $B$.
Then, the bug has a $\frac{2}{3}$ chance of reaching a new vertex next. Call this $C$. $A, B,$ and $C$ are always on the same plane.
Now, we split cases.
Case $1$: The bug goes to the vertex $E$ opposite $A$ on the space diagonal with probability $\frac{1}{3}$.
Then, the bug has to visit $D$ on the plane of $ABC$ last, as there is no way in and out from $D$.
Therefore, there is only $1$ way out of $81$ to get to $D$ last.
Therefore, there is a $\frac{1}{243} \cdot \frac{6}{9} = \frac{6}{2187}$ chance of finding a good path in this case.
Case $2$: The bug goes to vertex $D$ on plane $ABC$ with a chance of $\frac{1}{3}$.
The bug then has only $1$ way to go to a point $E$ on the opposite face, therefore having a $\frac{1}{3}$ probability.
Then, the bug has a choice of two vertices on the face opposite to $ABCD$.
This results in a $\frac{2}{3}$ probability of finding a good path to a point $F$.
Then, there is only $1$ way out of $9$ to visit both other vertices on that face in $2$ moves.
Multiply the probabilities for this case to get $\frac{2}{243} \cdot \frac{6}{9} = \frac{12}{2187}$.
Add the probabilities of these two cases together to get $\frac{18}{2187} = \boxed{\textbf{(C) }\frac{2}{243}}.$