Combinatorics is the mathematics of counting, selecting and arranging objects. Combinatorics include the theory of permutations and combinations. These topics have an enormous range of applications in pure and applied Mathematics and Computer Science. These are processes by which we organize sets so that we can interpret and apply the data they contain. Generally speaking, combinatorial questions ask whether a subset of a given set can be chosen and arranged in a way that conforms with certain constraints and, if so, in how many ways it can be done. Applications of combinatorics play a major role in the analysis of algorithms. For example, it is often necessary in such analysis to count the average number of times that a particular portion of an algorithm is executed over all possible data sets. This topic also includes solution of difference equations. Differences are required for analysis of algorithmic complexity and since computers are frequently used in the numerical solution of differential equations via their discretized versions which are difference equations. It also deals with questions about configurations of sets, families of finite sets that overlap according to some prescribed numerical or geometrical conditions. Skill in using combinatorial techniques is needed in almost every discipline where mathematics is applied. Salient Features Over 1000 problems are used to illustrate concepts, related to different topics and introduce applications. Over 1000 exercises in the text with many different types of questions posed. Precise mathematical language is used without excessive formalism and abstraction. Problem sets are started clearly and unambiguously and all are carefully graded for various levels of difficulty.
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