Derangements, a favorite topic in combinatorics, are usually studied using the inclusion-exclusion principle, to calculate the number of derangements of n objects, as well as the probability of a derangement occurring. This paper briefly presents this solution, as well as a second fairly standard solution using a recursion method, and then proceeds to solve for the probability of a derangement using the binomial inversion formula, which is derived in the final section of the paper. To show the utility and elegance of this approach, the expected value of correct assignments is also calculated if n objects are arranged at random.
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"The Old Hats Problem,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 4
, Article 2.
Available at: https://scholar.rose-hulman.edu/rhumj/vol4/iss1/2