A probability problem

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A Dartmouth student asked his math teacher Dana Williams if he could solve the following problem:

QUESTION: We start with n ropes and gather their 2n ends together.
Then we randomly pair the ends and make n joins. Let E(n)
be the expected number of loops. What is E(n)?

You might be interested in trying to solve this problem. You can check your answer here.


(1) There is probably a history to this problem. If you know a source for it please mention this on the discussion page above.

(2) Can you determine the distribution of the number of loops? If not estimate this by simulation and report you results on the discussion page.