This is related to the Coupon collector's problem as noted in the comments.

Building off of [this][1] 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][2].

  [1]: https://math.stackexchange.com/a/693254/697491
  [2]: https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Asymptotic_approximation