Problem E

Arranging Tiles 3

Eliana has a box of tiles, each with an integer from \(0\) to \(9\) on it. Each integer appears on at least three tiles. Eliana creates larger numbers by placing tiles side by side. For example, using the tiles \(3\) and \(7\), Eliana can create the \(2\)-digit number \(37\) or \(73\).

Using six of her tiles, Eliana forms two \(3\)-digit numbers, \(ABC\) and \(DEF\), that add to \(1234\).

Eliana then notices that \(A>D,~B>E,\) and \(C>F\). How many possible \(6\)-tuples \((A,~B,~C,~D,~E,~F)\) could she have chosen?