This is related to the Coupon collector's problem as noted in the comments.
Building off of this post, the probability of observing $k$ unique letters in $m$ random samples from an alphabet of size $n$ is:
$\big\{\!{m\!\atop{k}}\big\}\binom{n}{k}\frac{k!}{n^m}$
Where $\big\{\!{m\!\atop{k}}\big\}$ is the Stirling number of the second kind.
For large $m$, $\ln\big(\big\{\!{m\!\atop{k}}\big\}\big)$ can be approximated.