You roll a dice with $s$ sides for $n$ times. What is the probability that $t$ sides appeared once?

Example with $s=2$, $n=3$: you roll a dice with 2 sides for 3 times. possible outcomes: '1' and '2'.

- $p(t=0)$ is the probability that '1' appeared 0 times and '2' appeared 3 times + the probability that '2' appeared 0 times and '1' appeared 3 times, that is 1/4.
- $p(t=1)$ is the probability that '1' appeared once and '2' appeared twice + the probability that '2' appeared once and '1' appeared twice, that is 3/4.
- $p(t=2)=0$, because it is impossible that both '1' and '2' appear once.

If there is a formula for $p(s,n,t)$ it will be awesome, but also an algorithm is welcome.