I have seen a few related questions but not exactly what I am looking for (in particular [this][1] and [this][2]) I think. I may be missing something since I am only a beginner at these counting problems.

I am trying to calculate the amount of times (experiments) I need to perform were I choose $k$ items from a list of $n$ items so that I have a probability $P$ that each of the n items was selected at least once.

I attacked the problem by starting small and saying I have 5 items and each experiment consists of drawing two of them (without replacement). After the first experiments I reasoned that I have 100% probability that there are 3 items not selected yet. I proceeded to calculate the probability that after the second experiment I have three, two, or one items not selected. And so on with the third experiment and more.

Unfortunately, I am having trouble generalizing this approach to $n$ and $k$ and to a number of experiments $m$.

I suspect there is a good chance that this is a duplicate. If so can someone please point me in the right direction?

  [1]: https://stats.stackexchange.com/questions/30483/probability-of-selecting-each-item-at-least-once-when-sampling-with-replacement
  [2]: https://stats.stackexchange.com/questions/202313/mars-attack-probability-to-destroy-n-spaceships-with-k-cdot-n-missiles