In this paper the coupon collector's problem with group drawings is studied. Assume there are $ n $ different coupons. At each time precisely $ s $ of the $ n $ coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein's method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n={n\over s}(α\log(n)+x)$, where $0<α<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $α=1$, again using Stein's method.