Expected number of swaps in a derangement

Discrete Mathematics Level 4

6 cards numbered 1 through 6 are in a derangement. We define a random variable S to be the minimum number of pairs of cards that must be swapped so that the cards end up in increasing numerical order. If no card is in the position that it is to end up in, then the expected value of \(S\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

A derangement is a permutation of the elements, such that none of them are in their original position.

You may use the fact that there are 265 derangements on 6 elements.


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