A Bucket Of Shoelaces

Discrete Mathematics Level 5

Anton has a bucket of 5 shoelaces:

He randomly selects two distinct "ends" of a shoelace (from all ends -- there are 10 in the beginning) and ties them together to form a longer shoelace. Sometimes, he happens to choose ends of the same shoelace (or chain of shoelaces), and when he ties them together it forms a loop (which has no ends).

Eventually, he will only have loops (at least 1, and no more than 5). What is the expected value for the number of loops Anton will have in the end?

Give your answer to 3 decimal places.


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