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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Matching Problem

Suppose I return N homeworks randomly to my N students. What is the chance that no student gets back her own homework? In a class of 30? Would it be more or less if I had more students?

Surprising answer: the chance that no one gets back her own homework is approximately (1/e), which is approximately 36.8 percent of the time!

And the answer is about the same, no matter how many students you have! (It gets closer and closer to 1/e the larger N gets, and is already a very good approximation for 5 students.)

Gee, where did the number e come from?

Presentation Suggestions:
Poll the class about the probabilities before you tell them the answer, just to see how good their intuition is.

The Math Behind the Fact:
This interesting fact is usually proved in classes in discrete mathematics or probability, using something called the inclusion-exclusion principle. The answer turns out to be the partial sum of an infinite series for 1/e.

Su, Francis E., et al. "Matching Problem." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    probability, combinatorics
Subjects:    combinatorics, probability
Level:    Easy
Fun Fact suggested by:   Francis Su
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