Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back.

**$N=1$ is standard coupon collector problem. I have not seen any reference for $N>1$.**

**1.** What is the probability that after $k$ trials you selected each coupon from $1$ to $T$ at least once?

**2.** What is the expected number of trials one needs to select each coupon from $1$ to $T$ at least once with probability $1-\frac{1}{c}$ for some $c>1$?

**3.** Fix a coupon $x_t\in\{1,\dots,T\}$. How many trials to do you expect to have to select this particular coupon at least once with probability $1-\frac{1}{c}$ for some $c>1$?

I am looking for sharp asymptotics.