Say I have 9 uniquely colored balls. I want to select them into 3 different groups. Each grouping must contain at least 1 ball, and all the balls must be selected. The first 2 groupings are interchangeable and not distinct from each other, but are distinct from the 3rd group. The order within each group is also important.
E.g. using numbers for the balls: 12-34-56789 and 34-12-56789 are indistinguishable selections, but 56789-12-34 is distinct from those. 21-34-56789 is also different from 12-34-56789
How many different ways to do this are there?