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Suppose you have a K boxes, each with associated probability $p_i$. You start putting balls in the boxes with probability of putting a ball in a box being proportional to $(n_i+1)\cdot p_i$ where $n_i$ is the number of balls already in the box. Is there a closed form for the pmf for the number of balls in each of the boxes as a function of the total number of balls and if so, what's this distribution called?

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