I originally posted this question here:

https://math.stackexchange.com/questions/1296199/combinatorial-formula-for-the-number-of-different-words :

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 $d\ll m.$

wordsin the title asworlds. Too bad. $\endgroup$off the wall. Your q. is harder than I thought at first. Thus I up-voted it. Now I expect the specialists to answer your question, possibly using some Bernoulli numbers or similar--let me see (let them sweat :-). $\endgroup$2more comments