Further questions about probability: the hunter and the rabbit The question is stated as follows:
Suppose there are $m$ hunters and $n$ rabbits. Each hunter shoots one out of the $n$ rabbits uniformly at random independently. Let X be the number of rabbits that are shot.
*we can assume that there are m shots in total,i.e each hunter take exactly one shot at one rabbit

*

*Suppose $r$ a non-negative integer. Compute $\text{Prob}[X = r]$ in terms of $m$ and $n$.


*Compute $\mathbb E[X]$ in terms of $m$ and $n$.



*

*There are similar questions asked. But the ones I stated has a subtle difference since we are asked to give a probability of X = $r$ and calculate Expected Value.

My approach to this problem is as follows:
Since each hunter will shoot a rabbit at one time, and that two hunters may shoot the same rabbit, the total ways of shooting r rabbit is $r^m - C(n,1) \times r^{m-1} + C(n,2) \times r^{m-2} - ... $ (Inclusion-Exclusion Principle) and the total shootings are $r^m$. And hence we derive the probability.
My question is:

*

*Is my approach correct?


*The expected values are a bit tricky. One way is to use the probability I calculated above and multiply each by r. Another way is to set another set of random variables.(Supposingly makes it easier, but I don't think it can be used here since, though the hunter shoots independently, the rabbit doesn't die independently)
How to calculate the expected value here?
 A: Various questions relating to this problem have been asked before on math.stack.exchange. Here is an answer to part $2$ of your question.  Note, however, that in this answer, $\ m\ $ is the number of rabbits, not the number of hunters, and $\ n\ $ is the number of hunters.
There doesn't appear to be an answer giving the distribution of the number of rabbits killed for general values of $\ m\ $ and $\ n\ $.   While the principle of inclusion-exclusion is certainly the way to go to calculate this distribution, I don't understand how you arrived at your formula for the "number of ways of shooting $\ r\ $ rabbits". Also, the total number of possible ways the rabbits can be shot is $\ n^m\ $, not $\ r^m\ $.  This just the number of functions from the set of $\ m\ $ hunters to the set of $\ n\ $ rabbits.
The total number of ways that a specific set of $\ r\ $ rabbits can be shot is just the number of surjective functions from the set of hunters onto that set of rabbits.  By the principle of inclusion-exclusion, this is
\begin{align}
\cases{\displaystyle\sum_{i=0}^{r-1}(-1)^iC^r_i(r-i)^m&if $ r\le m\ $, or\\
0&if $\ r>m\ $.}
\end{align}
However, there are $\ C^n_r\ $ distinct sets of $\ r\ $ rabbits, so the total number of ways the hunters can shoot exactly $\ r\ $ rabbits is
$$
C^n_r\sum_{i=0}^{r-1}(-1)^iC^r_i(r-i)^m\ ,
$$
for $\ r\le m\ $, and the probability of their doing so is therefore
$$
\text{Prob}[X=r]=\frac{\displaystyle C^n_r\sum_{i=0}^{r-1}(-1)^iC^r_i(r-i)^m}{n^m}\ .
$$
