This book is mostly based on the author’s 25 years of teaching combinatorics to two distinct sets of students: first-year students and seniors from all backgrounds, not just limited to only those majoring in mathematics and physics. The prerequisites are kept to a minimum; essentially, only high school algebra is required. The design is to go from zero knowledge to advanced themes and various applications during a semester of three or three and a half months with quite a few topics intended for research projects and additional reading.This unique book features the key themes of classical introductory combinatorics, modeling (mainly linear), and elementary number theory with a constant focus on applications in statistics, physics, biology, economics, and computer science. These applications include dimers, random walks, binomial and Poisson distributions, games of chance (lottery, dice, poker, roulette), pricing options, population growth, tree growth, modeling epidemic spread, invasion ecology, fission reactors, and networks.A lot of material is provided in the form of relatively self-contained problems, about 135, and exercises, about 270, which are almost always with hints and answers. A systematic introduction to number theory (with complete justifications) is a significant part of the book, including finite fields, Pell’s equations, continued fractions, quadratic reciprocity, the Frobenius coin problem, Pisano periods, applications to magic and Latin squares and elements of cryptography. The recurrence relations and modeling play a very significant role, including the usage of Bessel functions for motivated readers. The book contains a lot of history of mathematics and recreational mathematics.Key FeaturesA unique textbook devoted to introductory combinatorics, modeling and elementary number theoryIt is relatively simple to use in the classroom: 1 Chapter is designed for 2 weeks, and the notes can be provided designed for remote teaching or to display in the classroomSome of the advanced topics are good for research projects; the most popular ones in the author’s classes were dimers, modeling tree growth, Magic squares and Pisano periods via finite fieldsElementary number theory is exposed systematically (with complete proofs and good ones)An impressive collection of about 135 problems and 270 exercises (mostly with hints and answers)It contains a lot of history of mathematics and the games of chance, and many topics traditional in recreational mathematicsThe book also serves motivated high-school students, teachers of mathematics, and specialists in neighboring fields interested in combinatorics and its applications

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