Coupon collector: expected coupons drawn until $N$ of some coupon have been collected?

Question

I'm interested in determining the expected number of coupons drawn out of a universe of $C$ distinct coupon types such that $N$ of some coupon have been drawn.

I've used a Markov Chain successfully for small cases (small $C$ and $N$), but that method becomes infeasible obviously for larger cases.

There is a question and answer dealing with $N=2$, but I don't see how that might be extended for $N>2$.

Is there a direct way to arrive at the desired result?

Answer 1

For $n$ coupon types, and desried quantity $k$ of some coupon,

$\int_0^{\infty } e^{-t} \left(\frac{e^{t/n} \space\space \Gamma\left(k,\frac{t}{n}\right)}{\Gamma (k)}\right)^n \, dt$

will produce the desired result of expected draws to some coupon having $k$ copies.