A complete set of solutions in integers \(n\geq 2\) and \(m\geq 4\) is given for the binomial diophantine equation \[ {n\choose 2}={m\choose 4}.\tag{1} \] Setting \(u=2n-1\), \(v=2m-3\), equation (1) takes the form \[ 48u^2=v^4- 10v^2+57,\tag{2} \] which shows that (1) represents an elliptic curve. In order to solve (1) it clearly suffices to solve equation (2) in \(u\) and \(v\). The latter is now reduced to a pair of quartic Thue equations. On applying a transcendence result by \textit{A. Baker} and \textit{G. Wüstholz} [J. Reine Angew. Math. 442, 19-62 (1993; Zbl 0788.11026)] the author obtains absolute upper bounds for \(u\) and \(v\), which are subsequently reduced to a manageable size by an LLL-reduction process.