[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Prove the following binomial identity

[[math]] {2n \choose n} = \sum_{j = 0}^n { n \choose j}^2\ . [[/math]]

Hint: Consider an urn with [math]n[/math] red balls and [math]n[/math] blue balls inside. Show that each side of the equation equals the number of ways to choose [math]n[/math] balls from the urn.