Railway routes

The diagram below represents a simplified railway system, and we want to know how many different ways there are of going from A to E, if we never go twice along the same line in any journey.

This is a very simple proposition, but practically impossible to solve until you have hit on some method of recording the routes. You see there are many ways of going, from the short route ABDE, taking one of the large loops, up to the long route ABCDBCDBCDE, which takes you over every line on the system and can itself be varied in order in many ways. How many different ways of going are there ?

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