Probability of finding $m$ or more cells empty I asked I question here trying to obtain clarification about how to follow a hint. In spite of the fine answers I received there, the hint doesn't look very helpful. I'd like to know a hint for the following problem or a way to use the hint I already have.
The probability $p_{m}(r,n)$ of finding exactly $m$ cells empty placing $r$ balls into $n$ cells is $$p_{m}(r,n)=\frac{1}{n^{r}}\binom{n}{m}A(r,n-m)=\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{\nu}\left(1-\frac{m+\nu}{n}\right)^{r}$$
From such probability, conclude that the probability $x_{m}(r,n)$ of finding $m$ or more cells empty equals
$$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{\nu}\left(1-\frac{m+\nu}{n}\right)^{r}\frac{m}{m+v}$$
HINT: Evaluate $x_{m}(r,n)-p_{m}(r,n)$.
Without using the hint, I tried to find the pattern for $p_{m}$, $p_{m+1}$, etc with the goal of factoring each term and sum over something recognizable, but I couldn't find a way to factor the last term (for example, to factor $(n-m-1-\nu)^{r}$ into some terms including the original expression $(n-m-\nu)^{r}$...I don't think it is possible. Furthermore, I don't think that strategy is going to produce the answer I'm looking for.
 A: See the following (Theorem 2). This gives the probability that exactly $m$ cells are empty. You could probably modify this to get the multiplicative factor $\frac{m}{m+v}$.
A: To use Feller's original notation. Let $E_m(r,n)$ be the number of distributions leaving exactly $m$ cells empty, then we have $$p_m(r,n)=n^{-r}\binom{n}{k}A(r,n-m)$$
as the corresponding probability ($A$ is to be defined)
Now, we need compute the probability $x_m(r,n)$ of finding $m$ or more cells empty. 
Now, the probability that $m$ or more cells are empty is given by
$x_m = p_m + p_{m+1} + \cdots + p_{n}$. Now, observe the reason why the hint was given: $x_{m+1} = x_m - p_m$ (i.e., $m+1$ or more cells being empty is the same as $\ge m$ cells being empty minus exactly $m$ cells being empty).
So for a sanity test, you could plug in the value of $x_m(r,n)$ given, and verify whether $x_{m}(r,n)-x_{m+1}(r,n)=p_m(r,n)$ holds (with the appropriate boundary cases).
The deeper underlying reason for this hint is that Feller probably wants you to figure out an inclusion-exclusion type argument. If I get time, I will write out the details. But the above should be a sufficiently detailed hint?
