I originally posted this question here:
I am interested in the asymptotic behaviour of the following quantity:
Suppose we have $m$ distinct letters and we are allowed to use each letter at most $d$ times. What is the number of distinct words of length $k$ that can be formed?
Indeed, one can find a recurrence formula, but I do not quite see how one can find a uniform asymptotic for all $m,d,k.$
Edit: After discussion in the comments, I can reduce my problem to the range, $m\ge k$ and $d\ll m.$