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Article Title

The Old Hats Problem

Abstract

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.

Author Bio

I am currently a high school sophomore at the Westridge School for Girls in Pasadena, California. I have loved math for as longas I can remember, and hope to be enjoy its beauty always. I became interested in combinatorics as a freshman, and started investigating derangements using different techniques. With my advisor, Dr. John Woo, we developed a binomial inversion approach to tackling problems of derangments, and presented this work at the Spring Southern California meeting of the MAA at Caltech in 2002, where I was the lone high schooler presenting. That was very intimidating, but also thrilling.

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