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A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

enter image description here

The answer is 2400

What is a way to do this combinatorics problem that could generalize to do any of problems similar to this but with more path?

i have limited terminology knowlegde, i would like a technical solution but please explain

No answer for a week already, Can anybody gave some clues?

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    @GerryMyerson - I don't mean to avoid casework but think of a method that make the solution divide into minimize but same number of cases, that is to make the calculation robotic2012-03-06

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