413. Railway Routes. The diagram shows a simplified railway network. We want to find out how many different ways there are to travel from \(A\) to \(E\), without passing through the same segment more than once on any route.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-140.jpg?height=477&width=962&top_left_y=1786&top_left_x=547)

The question is very simple. However, it is practically impossible to answer it until you come up with a method to record all the routes. The issue is that there are too many routes, ranging from the short \(A B D E\), which includes one large arc, to the long \(A B C D B C D B C D E\), which includes every segment of our system and allows for various variations.

How many different routes are there in total?
413. There are 2501 routes from \(B\) to \(D\), specifically:

\begin{tabular}{cccr}
\begin{tabular}{c} 
Number of \\
segments
\end{tabular} & \begin{tabular}{c} 
Number of \\
variations
\end{tabular} & \begin{tabular}{c} 
Number of \\
routes
\end{tabular} & \\
1 & 1 & 2 & 2 \\
2 & 1 & 9 & 9 \\
3 & 2 & 12 & 24 \\
4 & 5 & 18 & 90 \\
5 & 4 & 72 & 288 \\
6 & 14 & 36 & 504 \\
8 & 22 & 72 & 1584 \\
\({ } &{ } &{2501}\)
\end{tabular}

It is sufficient to consider the routes from \(B\) to \(D\). A route consisting of 1 segment leads directly to \(D\). The route with 2 segments is \(C D\). The routes with 3 segments are \(C B D\) and \(D C D\). The five routes with 4 segments are \(D B C D, D C B D, C B C D, C D C D\) and \(C D B D\). Each of these routes has variations associated with
the choice of specific segments, and the number of such variations is the same for any route containing the same number of segments. There are no routes with seven segments.