Probability of $x$ non-zero multinomial coefficients? Let $\prod_{k=1}^K p_k^{c_k}$ be the joint probability distribution of some $n$-long memoryless random process (thus, $\sum_{k=1}^Kp_k=1$), where each $c_k$ tells how many times the $k$-th element appeared. Thus, the vector $c=(c_1,...,c_K)$ follows the multinomial distribution with $\sum_{k=1}^Kc_k=n$.

What is the probability that $x\in \mathbb{N}$ entries in the vector $c$ are non-zero?

Or analogously, what is the probability that $y$-entries are equal to zero?
Happened to come across this. Think I solved that for this article (for a uniform distribution of the p_k):
https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1444673?src=recsys&amp;journalCode=utas20
Happened to come across this. Think I solved that for this article (for a uniform distribution of the p_k):
https://www.tandfonline.com/doi/abs/10.1080/00031305.2018.1444673?src=recsys&amp;journalCode=utas20