A fundamental question in social choice and multi-agent systems is aggregating ordinal preferences expressed by agents into a measurably prudent collective choice. A promising line of recent work views ordinal preferences as a proxy for underlying cardinal preferences. It aims to optimize distortion, the worst-case approximation ratio of the (utilitarian) social welfare. When agents rank the set of alternatives, prior work identifies near-optimal voting rules for selecting one or more alternatives. However, ranking all the alternatives is prohibitive when there are many alternatives. In this work, we consider the setting where each agent ranks only her t favorite alternatives and identify almost tight bounds on the best possible distortion when selecting a single alternative or a committee of alternatives of a given size k. Our results also extend to approximating higher moments of social welfare. Along the way, we close a gap left open in prior work by identifying asymptotically tight distortion bounds for committee selection given full rankings.