This multivolume work on the analysis of algorithms has long been recognized as the definitive description of classical computer science. The four volumes published to date already comprise a unique and invaluable resource in programming theory and practice. Countless readers have spoken about the profound personal influence of Knuth’s writings. Scientists have marveled at the beauty and elegance of his analysis, while practicing programmers have successfully applied his “cookbook” solutions to their day-to-day problems. All have admired Knuth for the breadth, clarity, accuracy, and good humor found in his books. To continue the fourth and later volumes of the set, and to update parts of the existing volumes, Knuth has created a series of small books called fascicles, which are published at regular intervals. Each fascicle encompasses a section or more of wholly new or revised material. Ultimately, the content of these fascicles will be rolled up into the comprehensive, final versions of each volume, and the enormous undertaking that began in 1962 will be complete. This fascicle covers three separate topics: Mathematical Preliminaries.  Knuth writes that this portion of fascicle 5 "extends the ‘Mathematical Preliminaries’ of Section 1.2 in Volume 1 to things that I didn't know about in the 1960s. Most of this new material deals with probabilities and expectations of random events; there's also an introduction to the theory of martingales." Backtracking: this section is the counterpart to section 7.2.1 which covered the generation of basic combinatorial patterns.  This section covers non-basic patterns, ones where the developer needs to make tentative choices and then may need to backtrack when those choices need revision. Dancing Links: this section is related to 2 above.  It develops an important data structure technique that is suitable for backtrack programming described above.