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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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    What's your question?2012-02-25
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    @Alex: I believe that Victor is looking for a general solution for digraphs of this type.2012-02-25
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    @BrianM.Scott The last sentence was not present when I inquired. It clears things up nicely.2012-02-25
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    @BrianM.Scott - exactly, however i have limited terminology knowlegde, i would like a technical solution but please explain2012-02-25
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    How do you know the answer is 2400? Do you have a worked-out solution of this problem? Does it come from some source that might give some general ideas?2012-02-27
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    The problem has also been posted to http://www.artofproblemsolving.com/Wiki/index.php?title=2012_AMC_10B_Problems/Problem_25&oldid=45128 where it is identified as 2012 AMC 10B Problems/Problem 25.2012-02-27
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    No answer for a week already to my question, how do you know the answer is 2400. Can **you** give some clues?2012-03-06
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    @GerryMyerson - it is in here but i don't like casework becauese it is not always work for bigger case: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=4&cts=1331076718238&ved=0CEMQFjAD&url=http%3A%2F%2Fwww.gliyanet.com%2Fmath%2Fcontests%2FAMC%2F2012AMC10BSolutions.pdf&ei=a55WT5ifAYK40QGit8GmCg&usg=AFQjCNGDoaqG0IgjAClM5UZbuSXc3125GA2012-03-06
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    I think the problem is inherently difficult and the solution you have linked to may be the simplest way to do it. In particular, I don't see any way to avoid casework. Maybe someone else will.2012-03-06
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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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