# 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.

DISCUSSION QUESTION:

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

Can you determine the distribution of the number of loops? If not estimate this by simulation.