#### Vol. 7, No. 5, 2014

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The sock matching problem

### Sarah Gilliand, Charles Johnson, Sam Rush and Deborah Wood

Vol. 7 (2014), No. 5, 691–697
##### Abstract

When matching socks after doing the laundry, how many unmatched socks can appear in the process of drawing one sock at a time from the basket? By connecting the problem of sock matching to the Catalan numbers, we give the probability that $k$ unmatched socks appear. We also show that, for each fixed $k$, this probability approaches $1$ as the number of socks becomes large enough. The relation between the number of socks and the $k$ for which a given probability is first reached is also discussed, but a complete answer is open.

##### Keywords
Catalan numbers, sock matching, Dyck paths
##### Mathematical Subject Classification 2010
Primary: 05A15, 05A16
Secondary: 03B48, 00A69
##### Milestones
Received: 21 August 2013
Accepted: 25 November 2013
Published: 1 August 2014

Communicated by Jim Haglund
##### Authors
 Sarah Gilliand Department of Biology The College of William & Mary College Station Unit 3011 P.O. Box 8793 Williamsburg, VA 23187 United States Charles Johnson Department of Mathematics The College of William & Mary P.O. Box 8795 Williamsburg, VA 23187 United States Sam Rush Department of Computer Science California Institute of Technology 1200 East California Boulevard, MS 305-16 Pasadena, CA 91125 United States Deborah Wood Department of Mathematics The College of William & Mary College Station Unit 4085 P.O. Box 8793 Williamsburg, VA 23187 United States