Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1. Starting with no edges between A and B, if |E| many edges are added between A and B uniformly at random, what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a potential perfect matching, let X sub S = 1 if all the edges in S are added to the graph, and X sub S= 0 if any of them are missing. What isE[X sub S]?

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