Expectations Between Jokers

A standard \(52\)-card deck with \(n\) extra jokers (where \(n\ge 2\)) is shuffled. Given the expected number of cards between the first joker and the last joker exclusive is a positive integer \(E\), find the sum of all possible values of \(E\).

Details and Assumptions

The \(52\)-card deck with \(n\) extra jokers implies that there are \(52+n\) cards in total.

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