We propose a demand model where consumers simultaneously choose a few diﬀerent goods from a large menu of available goods, and choose how much to consume of each good. The model nests multinomial discrete choice and continuous demand systems as special cases. Goods can be substitutes or complements. Random coeﬃcients are employed to capture the wide variation in the composition of consumption baskets. Non-negativity constraints produce corners that account for diﬀerent consumers purchasing diﬀerent numbers of types of goods. We show semiparametric identiﬁcation of the model. We apply the model to the demand for fruit in the United Kingdom. We estimate the model’s parameters using UK scanner data for 2008 from the Kantar World Panel. Using our parameter estimates, we estimate a matrix of demand elasticities for 27 categories of fruit and analyze a range of tax and policy change scenarios.