Combination from arbitrary number of type of items

Question

Suppose we have $K$ number of items and the probability of picking item $k\in \{1,\dots,K\}$ equals $p_k$ (with $\sum_{k=1}^K p_k =1$). Let $n$ be an arbitrary number, then what is the probability that for each $k$ we have picked exactly $a_k$ items, with $\sum_{k=1}^K a_k = n$? Is this some known quantity like $\binom{n}{k}$?

Answer 1

Assuming that $n$ is the number of trials (picked items) the probability in question is: $$ n!\prod_{k=1}^K\frac{p_k^{a_k}}{a_k!}. $$