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.

How to Cite this Page:
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 Suggestions? Use this form. 4.02 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! New: get the MathFeed iPhone App! Brings you news and views on math: showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!