You have 22 numbers. You sample from them with replacement. You want to know the expected number of times you have to draw a number, before you end up drawing each of them at least once. Is that it? Or are you asking something else?
Yes that is exactly what I am looking for.
Well, I honestly don't see how the problem you keep giving us is the necessarily the same as the one I gave you just now, but I'll start talking about the problem that I posed for you.
We generally don't give homework answers
here (I guess?), but I'll tell you what my approach was:
(1) Start by solving this problem instead: "You have n numbers. You sample from them with replacement. What is the expected number of times you have to draw before you end up drawing each of them at least once?"
(2) Let's give a name to the answer: call it e(n). So e(1) = 1. e(2) = 3, because it takes you one draw to come up with the first distinct number, and from that point it should take you 2 draws on average to get a second distinct number.
(3) Try to find a formula for e(n) in terms of e(n-1). (That's called a "recursive formula" for e(n).)
(4) Guess the formula for e(n) in terms of n, just by looking at the recursive formula, and figuring out creative ways of writing e(3), e(4), etc. If I'm right, the answer turns out to be n multiplied by the nth partial sum of a well-known infinite series.
There's probably a clever, more logical way of doing the problem. But that was my own "hack" approach.