The main result of the paper is a proof of the equivalence of single and multiple lottery mechanisms for the problem of allocating students to schools in which students have strict preferences and the schools are indifferent. This solves a recent open problem proposed by Pathak, who was motivated by the practical problem of assigning students to high schools in New York City. In proving this result, a new approach is introduced, that simplifies and unifies all the known equivalence results in the house allocation literature. Along the way, two new mechanisms---Partitioned Random Priority and Partitioned Random Endowment---are introduced for the house allocation problem. These mechanisms generalize several known and well-studied mechanisms for the house allocation problem and are particularly appropriate for many-to-one versions of the problem, the school choice problem being the most prominent.