Let's say we have an expanding system of slots(you can be keep increasing the number of slots). Now I have a very large number of coins in a bag, all equi-similar, labeled(1,2,..n). Now the experiment is, you take a random coin from the bag and put it in a slot. Now you keep repeating this until you have at-least one coin of each label.
Variant: You keep repeating until you have the sequence(1,2,..n) in the slots, by sequence I mean the coins need not be adjacent.
What is the expected number of slots we would need?
Edit:
I didn't want to include this, but since MSE suggested my question didn't have enought context,
Basically my goal is to calculate the expected number of visits to some n states given that, as soon as all the states are visited in order 1,2...n(again this needs to appear only as a sequence in the visit order) the visits end.