If I roll an n-sided die m times, what is the probability of getting only one face, two faces, .... or n faces?
Let us assume that I have a 6-sided die and I roll it 10 times. I want to calculate different probabilities. The probability of getting only one face from all rolls. The probability of getting only 2 faces in all rolls and so on until calculating the probability of the appearance of the 6 faces (at least one time for each face) in all rolls.
The probability that $x$ faces have appeared at least once after $m$ throws of a fair $n$-faced die is $$Pr(X=x \mid m) = \frac{S_2(m,x) \, n!} {(n-x)!\,n^m}$$ where $S_2(m,x)$ is a Stirling number of the second kind
So for example with $n=6$ and $m=10$ you would get $$Pr(X=2) = \frac{511 \times 720} {24 \times 60466176} \approx 0.000254$$
You could calculate this recursively using $$Pr(X=x \mid m)=\frac{n-x+1}{n}Pr(X=x-1 \mid m-1) + \frac{x}{n}Pr(X=x \mid m-1)$$ starting with $Pr(X=0 \mid m)=0$ and $Pr(X=x \mid 0)$ except $Pr(X=0 \mid 0)=1$. That would give a table like this:
x 1 2 3 4 5 6
m
1 1 0 0 0 0 0
2 0.166667 0.833333 0 0 0 0
3 0.027778 0.416667 0.555556 0 0 0
4 0.004630 0.162037 0.555556 0.277778 0 0
5 0.000772 0.057870 0.385802 0.462963 0.092593 0
6 0.000129 0.019933 0.231481 0.501543 0.231481 0.015432
7 0.000021 0.006752 0.129029 0.450103 0.360082 0.054012
8 0.000004 0.002268 0.069016 0.364583 0.450103 0.114026
9 0.000001 0.000759 0.036020 0.277563 0.496614 0.189043
10 0.0000001 0.000254 0.018516 0.203052 0.506366 0.271812
