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A perfect matching is a matching of all the vertices of a graph. In other words, a perfect matching is a set of edges such that each vertex of the graph is incident to exactly one edge in the set.

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Matching with probabilistic edges

Linearity of expectation might be helpful here. Pick any pairing of vertices, the probability that all of these edges exist is $p^{n}$. So the expected number of matchings must be $n!p^{n}\simeq (\f …
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