Counting the Paths Backprop Skips

A network is laid out as a computational graph:

• one input node $x$;
• 10 hidden layers, each containing exactly 2 nodes;
• both nodes of layer $1$ take $x$ as input, and every node in layer $k$ takes both nodes of layer $k-1$ as inputs;
• one output node $L$, which takes both nodes of layer $10$.

One way to get $\dfrac{\partial L}{\partial x}$ is to apply "multiply along each path, then add across paths" literally: list every distinct path from $x$ to $L$, multiply the local derivatives along it, and sum the results. Backpropagation avoids this by visiting each edge just once.

How many distinct paths from $x$ to $L$ would the path-by-path method have to sum over?