Upper bound for $n$ balls in $m$ bins

Question

When placing k balls randomly into k bins we can bound the number of balls that falls into any bin by

$$p \leq \frac{3\ln(k)}{\ln(\ln(k))}$$

and this bound fails to hold with probability $\leq$ $\frac1k$

However, what if we have n balls and m bins where $n \neq m$ ? Is there a way to bound this as well?