We introduce Label-Combination Prototypical Networks (LC-Protonets) to address the problem of multi-label few-shot classification, where a model must generalize to new classes based on only a few available examples. Extending Prototypical Networks, LC-Protonets generate one prototype per label combination, derived from the power set of labels present in the limited training items, rather than one prototype per label. Our method is applied to automatic audio tagging across diverse music datasets, covering various cultures and including both modern and traditional music, and is evaluated against existing approaches in the literature. The results demonstrate a significant performance improvement in almost all domains and training setups when using LC-Protonets for multi-label classification. In addition to training a few-shot learning model from scratch, we explore the use of a pre-trained model, obtained via supervised learning, to embed items in the feature space. Fine-tuning improves the generalization ability of all methods, yet LC-Protonets achieve high-level performance even without fine-tuning, in contrast to the comparative approaches. We finally analyze the scalability of the proposed method, providing detailed quantitative metrics from our experiments. The implementation and experimental setup are made publicly available, offering a benchmark for future research.