BALTIC-WAY 2011 Problem 7

Let $T$ denote the $15$ -element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$ .