How many times do I need to draw $$ items from a set on $$ items to get a probability $$ that each item is selected at least $$ times?

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This and this are related (but slightly different) questions. My question is more general.

I have a set of $N$ elements. I want to repeatedly draw $X$ items ($X \le N$) with replacement, i.e. I first choose $X$ items without replacement and then I replace them in the original set. I want each element to be drawn at least $K$ times. So how many times do I have to draw $X$ elements such that I have probability $P$ that each element has been drawn at least $K$ times?

Let's do an example. I have a deck of $N=10$ cards, labeled $1, 2, ..., 10$. At each step I draw $X=3$ cards, I look at them and then I put them back in the deck. How many times do I have to perform such step in order to draw each single card at least $K=5$ times with probability $P=0.95$?

EDIT: I would be happy also with a solution where $X=1$.

This is a double variation of the coupon collector's problem (which has $X=1,K=1$), and I suspect there is no easy calculation. The answer for a particular case may be easier to find by simulation; for example with your $N=10, X=3, K=5$ the expected number of draws seems to be about $28.3$, slight less than a third of the $X=1$ case which seems to be about $89.5$. To get $P=95%$ chance with $N=10, X=3, K=5$ my simulation suggest you seem to need about $38$ draws, again just under a third of the $X=1$ which seems to need about $122$ draws. — Henry, Nov 14 '20 at 11:24

How many times do I need to draw $$ items from a set on $$ items to get a probability $$ that each item is selected at least $$ times?

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