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How would you prove that the number of cuts in a graph (where cut is a set of edges which split two vertices) cannot be smaller than the number of directed paths from one vertex to the other?

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That would depend on what you know about graphs. Do you know the Max-Flow-Min-Cut theorem?

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Sure, but neither flow nor edge value is relevant. –  Ondrej Sotolar Dec 5 '11 at 0:51
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@Ondrej: They are if you assign each edge a capacity of $1$. Specifically, you want to look at Menger’s theorem; you may find the last two PDF’s listed under External links useful. –  Brian M. Scott Jan 4 '12 at 6:06

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