2012 AMC 10B Problem 25

Classify a path by the set $S$ of backward arrows it uses. If $S=\varnothing$, the path is determined by choosing one forward arrow in each column, giving $2\cdot2\cdot4\cdot4\cdot4\cdot2\cdot2=2^{10}$ paths. If $S$ uses only the left backward arrow, there are $2^8$ paths, and by symmetry the same for only the right backward arrow. If it uses both outer backward arrows but not the middle one, there are $2^6$ paths. If $S$ uses only the middle backward arrow, there are $2^9$ paths. If it uses the middle arrow and exactly one outer backward arrow, there are $2^7$ paths for each choice of outer arrow. If it uses all three backward arrows, there are $2^5$ paths. The total is $2^{10}$ $+2\cdot2^8$ $+2^6$ $+2^9$ $+2\cdot2^7$ $+2^5=2400$. Thus, E is the correct answer.