This work focuses on the development of randomized truthful mechanisms that seek to maximize both the revenue and the economic efficiency, or social welfare, of unit-demand auctions. In a unit-demand auction a set of k items is auctioned to a set of n consumers and no consumer can purchase more than one item. A framework is presented for devising polynomial-time randomized truthful mechanisms that are based on a new variant of the Vickrey-Clarke-Groves (VCG) mechanism. Instead of using reserve prices, this variant of VCG uses the number of objects that we wish to sell as a parameter. The mechanisms obtained differ er from each other in the way they select the number of items to be sold and allow an interesting trade-off between revenue and economic effciency, while improving upon the stateof- the-art results for the Unit-Demand Auction problem (09).