I'm trying to solve the following problem:
The probability $p_{i}$ that a given cell contains exactly i balls is given by the binomial distribution $p_{i}=\displaystyle \frac{\binom{r}{i}(n-1)^{r-i}}{n^{r}}$. Prove that the most probable number is the integer $v$ such that $\displaystyle \frac{(r-n+1)}{n}< v \leq \displaystyle \frac{(r+1)}{n}$.
I've tried to get the inequality using first $\displaystyle \frac{p_{i}}{p_{i-1}}$ (which didn't work quite well) and later $\displaystyle \frac{p_{i}}{p_{i+1}}$. For example, for the first attempt the result I obtained was:
$$\frac{p_{i}}{p_{i-1}}=\frac{r-i+1}{i(n-1)}$$
In this kind of problem, I'm expecting to obtain a $1$ in the resulting expression in order to figure out for which values of $i$ there is a maximum in the function, however, I can't get the correct result.
Any help will be appreciated.