High-quality labels are often very scarce, whereas unlabeled data with inferred weak labels occurs more naturally. In many cases, these weak labels dictate the frequency of each respective class over a set of instances. In this paper, we develop a unified approach to learning from such weakly-labeled data, which we call count-based weakly-supervised learning. At the heart of our approach is the ability to compute the probability of exactly k out of n outputs being set to true. This computation is differentiable, exact, and efficient. Building upon the previous computation, we derive a count loss penalizing the model for deviations in its distribution from an arithmetic constraint defined over label counts. We evaluate our approach on three common weakly-supervised learning paradigms and observe that our proposed approach achieves state-of-the-art or highly competitive results across all three of the paradigms.