Suppose I have $k$ bins to which I throw $k$ balls. The probability distribution over bins for each ball (independently) is some categorical distribution. i.e. there exist $p_1, \ldots , p_k$ that sum to 1 and describe the probability of falling into each bin.
What is the probability that after throwing $k$ balls there are no collisions? I think its $k!\cdot \prod_i p_i$ since there are $k!$ orders to achieve this, each with probability $\prod_i p_i$ and they are mutually exclusive. Is this analysis correct?