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This question already has an answer here:

Suppose I have a sticker album which consists of $N$ stickers. How many stickers should I buy in average to complete this album, assuming all the stickers appear with the same frequency?

More formally, let $X_1, X_2, \dots$ be independent random variables with an uniform distribution on the set $\{1, 2, \dots N\}$. Then, what is the expected value of the random variable $$ Z=\min {\left\{n\in\mathbb N: \forall i\leq N,\,\,\exists m\leq n:\, X_m=i\right\}} $$

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marked as duplicate by Henry probability Oct 17 '16 at 17:33

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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If you have $N-k$ and just need the last $k$, the probability that a sticker will be one you need is $k/N,$ so the expected number of stickers until that happens is $N/k$. Thus the total expected number of stickers is $$N \sum_{k=1}^N \dfrac1k$$

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