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I have a set of $n$ elements, $x_1,x_2,\dots,x_n$, each with a weight $w_1, w_2,\dots, w_n$. I want to sample a subset of fixed size $k$ such that the probability of drawing $\{x_a,\dots,x_b\}$ is proportional to $w_a\dots w_b$.

The only solution I could think of is some kind of MCMC, but I would prefer direct sampling because speed is of concern.

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