With the development of the digital economy, the power demand for data centers (DCs) is rising rapidly, which presents a challenge to the economic and low-carbon operation of the future distribution network. To this end, this paper fully considers the multiple flexibility of DC and its impact on the distribution network, and establishes a collaborative planning model of DC and distribution network. In this model, the interval method is utilized to capture the inherent uncertainties within the system (such as the renewable energy source (RES) output, wholesale market price, power load demand, carbon emission factor and workloads), and the planning model is transformed into a multi-objective optimization problem with interval parameters problem (IMOP) to minimize economic cost and carbon emission. On this basis, an interval multi-objective optimization evolutionary algorithm based on decomposition (IMOEA/D) is proposed to solve the IMOP and obtain the Pareto optimal solution while retaining all the uncertainty information. Finally, an improved IEEE 33-node distribution network is utilized as an example for simulation and analysis to confirm the efficacy of the proposed approach.