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*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by: **

Francis Su