Review: Mathematics Of Choice: Or, How To Count Without Counting

Mathematics Of Choice: Or, How To Count Without Counting by Ivan Morton Niven

My rating: 3 of 5 stars



This text is an engaging, even addictive, introduction to basic combinatorics. Written in a fun and inviting manner, reader interest is amplified by the author’s infectious enthusiasm. This is an excellent introduce to combinations and permutations. First published in 1975, before computers and calculators were assumed to be at the ready, the exercises in this book can all easily be done by hand on paper. Students finishing High School or in their first year of college will find this work an excellent adjunct to textbooks and lectures.

The work is arranged in a logical progression beginning with the definitions and motivations for factorials, combinations, and permutations. From there the reader moves to binomial coefficients, power sets, and Fibonacci numbers. The effect of repetitions on combinations makes a natural prelude in Chapter Four to the Inclusion-Exclusion Principle and the groundwork for basic probability. From partitions of integers the author moves into a brief and basic, yet cogent and enlightening, explanation of generating functions and some applications for them. The book also includes Pigeonhole Principle, induction, recursion, and allied topics.

Tom Schulte teaches mathematics at Oakland Community College in Michigan.


The William Lowell Putnam Mathematical Competition: Problems and Solutions 1965–1984 (MAA Problem Book Series)
Review by Tom Schulte

Unlike the rigorous and detailed prelude William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964, this work covers a roughly similar time span in one-fifth the space. The competition problems are displayed in a straightforward presentation, chronologically in the first section. The second section is the solutions, roughly three to five per page taking up the last four-fifths of the book.

Students participating in competition mathematics will benefit from these actual species of problem, seen “in the field.” However, anyone interested in problem solving at the advanced collegiate level will find the quality and diversity of the problems presented here both challenging and fun. Having the official solution safely several pages away allows one to solve first and compare later.

Winning teams and students from 1965 through 1984 are listed in an appendix. An index of problems by type, ranging from abstract algebra to Wallis product, allows the work to serve as a reference for challenging student questions on many topics. This compendium is a problem solver’s delight and the unadorned presentation compared to the previous volume in no way distracts from the historical interest and value of this potpourri of mathematics.



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