Suppose we have the hypergeometric distribution scenario, population size $N$, $K$ objects with a favorable feature. We draw $n$ times until $k$ successes but for each success we remove a $K$-type object from the population. We're increasing our chance of drawing an $(x+1)$-th object if we draw the $x$-th one by a bit by making the population smaller but decrease it by a another bit by removing one of the sought objects from the pile. Is this a known distribution? What's the PMF?
The original problem has multiple types of objects each with different weights assigned to them which point to Fisher's noncentral hypergeometric distribution so if you have the answer to the general problem too you're welcome to share it.
It $N$ denotes the number of trials needed for achieving $k$ successes then: $$N=X_1+\cdots+X_k$$ where the $X_i$ are independent and every $X_i$ has geometric distribution with parameter $p_i=\frac{K+1-i}{N+1-i}$.
In that sense the distribution is "known".
But finding its PMF seems to me quite a job.
I would only go after it if there are signals that it might be fruitful for some reason.