# Sample subsets with distribution proportional to elementwise product

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.