coupon collector - What is the probability of rolling $n$ dice until each side appears at least once?

We roll a die until each side appears at least once, then we stop.

What is the probability of rolling exactly $n$ dice?

I guess the answer is
$$6-6\left(\dfrac{5}{6}\right)^n\;,$$

but this may be wrong.

In trying to solve this problem, I read this paper, but I couldn't solve the problem.

Answer

I'm going to assume that what you are asking is: if we roll a die until all sides (i.e. faces) appear, what is the probability that we will roll $n$ times?

In order for this to happen, we need exactly five faces to have appeared in the first $n-1$ rolls, and then the one missing face must appear on the $n$-th roll. Using inclusion-exclusion, we can express this, for $n>1$, as

$$p(n) = \frac{1}{6} \binom{6}{5} \left( \left(\frac{5}{6} \right)^{n-1} - \binom{5}{4} \left( \frac{4}{6} \right)^{n-1} + \binom{5}{3} \left( \frac{3}{6}\right)^{n-1} - \binom{5}{2} \left( \frac{2}{6} \right)^{n-1}+\binom{5}{1} \left( \frac{1}{6}\right) ^{n-1} \right)$$
So, for example, $p(6)=5/324$, $p(7)=25/648$, and $p$ hits a maximum of $875/10368$ at $n=11$.

As a check, you can (numerically) verify that
$$\sum_{i=6}^{\infty} ip(i) = 14.7,$$
the well-known expected number of throws needed to see all faces.