From a deck of 52 cards, we draw $N$ times one card, with the return. Calculate the probability that each card in the deck will be drawn at least once.
I know that it looks a bit like Coupon collector's problem, but as I understand it, it's about expected value and variance. Here I need to look the other way around - I am given N and have to calculate the probability.
For N < 52 that will be 0 of course.
Then, for N = 52 we have: $1 \cdot \frac{51}{52} \frac{50}{52} \cdot ... \cdot \frac{1}{52}$ as I need to get a 'new card' each time and that probability is smaller with each draw as the set of 'old' (already chosen) cards gets bigger.
For $N = 53$ I have $1$ 'extra draw' that I can loose on drawing an 'old' card - one of $52$. But I can't just choose one of $51$ places between draws extending the set of already chosen cards and choose one of $52$ cards. I think that I would calculate many times the same sequences.
Therefore I lack the idea for scaling the idea from $52$ draws onwards. Any help would be appreciated.