Here is the question: Let $k,m,n$ be positive integers and $k\leq m\leq n$.
Compute $$\sum_{\substack{a_1+\dots+a_n=m,\\ 0\leq a_i<k, \text{for } i=1,2,\ldots,n}}\frac{m!}{a_1!a_2!\cdots a_n!}$$
The original question is to count the probability of the following event. Choose $m$ numbers $\{y_{i_1}, \ldots y_{i_m}\}$ from $n$ distinct numbers $\{y_1,\ldots,y_n\}$, we are allowed $y_{i_k}=y_{i_j}$.
What is the probability of the choice which has at least $k$ same numbers ?
If I am right, I think it only needs to compute the sum given above.
Thanks.