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New results related to maximizing the reliability of common systems with interchangeable redundancies at a component level have been obtained by using the method of algebraic inequalities. It is shown that for systems with independently working components with interchangeable redundancies, the system reliability corresponding to a symmetric arrangement of the redundant components is always inferior to the system reliability corresponding to an asymmetric arrangement of the redundant components, irrespective of the probabilities of failure of the different types of components. It is also shown that for series–parallel systems, the system reliability is maximized by arranging the main components in ascending order of their probabilities of failure, whereas the redundant components are arranged in descending order of their probabilities of failure. Finally, this article derives rigorously the highly counterintuitive result that if two components must be selected from n batches containing reliable and faulty components with unknown proportions, the likelihood that both components will be reliable is maximized by selecting both components from a randomly selected batch.