Distinct Orders to Break Stacked Targets

\leq i \leq r$) has $n_i$ targets stacked vertically. On each turn, the sniper selects a column that still has unbroken targets and shoots the lowest remaining target in that column. Once a column is empty, it can no longer be selected. In how many distinct orders can the sniper break all the targets? Solve this for $r = 5$ with $n_i = i$ (i.e., column 1 has 1 target, column 2 has 2, ..., column 5 has 5).