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Person Re-identification (ReID) aims at matching a person of interest across images. In convolutional neural network (CNN) based approaches, loss design plays a vital role in pulling closer features of the same identity and pushing far apart features of different identities. In recent years, triplet loss achieves superior performance and is predominant in ReID. However, triplet loss considers only three instances of two classes in per-query optimization (with an anchor sample as query) and it is actually equivalent to a two-class classification. There is a lack of loss design which enables the joint optimization of multiple instances (of multiple classes) within per-query optimization for person ReID. In this paper, we introduce a multi-class classification loss, i.e., N-tuple loss, to jointly consider multiple (N) instances for per-query optimization. This in fact aligns better with the ReID test/inference process, which conducts the ranking/comparisons among multiple instances. Furthermore, for more efficient multi-class classification, we propose a new meta prototypical N-tuple loss. With the multi-class classification incorporated, our model achieves the state-of-the-art performance on the benchmark person ReID datasets.