# Exercises in (Mathematical) Style: Stories of Binomial Coefficients

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*John McCleary*

MAA Press: An Imprint of the American Mathematical Society

Winner of a CHOICE Outstanding Academic Title Award for 2017!

Reprinted edition available:
NML/50

What does style mean in mathematics? Style is both how one does
something and how one communicates what was done. In this book, the
author investigates the worlds of the well-known numbers, the binomial
coefficients. The author follows the example of Raymond Queneau's
Exercises in Style. Offering the reader 99 stories in various
styles. The book celebrates the joy of mathematics and the joy of
writing mathematics by exploring the rich properties of this familiar
collection of numbers. For any one interested in mathematics, from
high school students on up.

#### Reviews & Endorsements

By examining and extending binomial coefficients from seemingly every possible direction, the author provides an amazing concoction of ideas, prompting readers to say "Wow, I forgot that connection," or "Wow, I did not know that," or just "Wow!..McCleary's effort is exceptional, as it reaches into the realm of élan, clearly demonstrating the energy and enthusiasm that can pervade mathematical writing and mathematics itself.

-- J. Johnson, CHOICE

# Table of Contents

## Exercises in (Mathematical) Style: Stories of Binomial Coefficients

- cover cover11
- Exercises in (Mathematical) Style iii4
- Preface vii8
- Contents xi12
- Notations 116
- Combinatorial thinking 318
- An algebraic relation 520
- Combinatorial consequences 722
- Downtown Carré 924
- Proof without words 1126
- A geometric representation 1227
- (6 choose 3) tetrads 1429
- A footnote 1530
- Fancy evaluations 1631
- Careful choices 1833
- An explicit formula 2035
- Explicit counting 2237
- History 2338
- The diagonal club 2540
- The Chu-Vandermonde identity 2742
- Up and down 2944
- Mind the gaps 3146
- Recurrence 3348
- Rabbits 3550
- Repetition 3752
- Left out 3954
- Sets 4156
- The Ramsey game 4358
- The Principle of Inclusion and Exclusion 4661
- Inversion 4964
- Derangements 5166
- Seating the visiting dignitaries 5469
- Multinomials, passively 5772
- Don't choose, distribute 5974
- Tanka** 6277
- Ode to a little theorem** 6378
- Divisibility by a prime** 6580
- A far finer gambit** 6883
- Congruence modulo a prime** 7186
- Alchemy** 7489
- Close reading** 7792
- An algorithm to recognize primes** 7994
- Shifting entries** 8196
- DIY primes** 8398
- Polynomial relations 86101
- The generalized binomial theorem 88103
- Four false starts 90105
- Protasis-apodosis** 93108
- Matrices** 96111
- Bourbaki 99114
- Generating repetitions 102117
- Dialogue concerning generating functions 104119
- Counting trees 107122
- q-analogues 109124
- A breakthrough** 112127
- Partitions of numbers 115130
- Take it to the limit 118133
- q-binomial theorem 120135
- The quantum plane*** 123138
- Scrapsheet 126141
- Mathematical Idol 129144
- Symmetries 132147
- Sums of powers* 135150
- Parts of proof 138153
- Rings and ideals*** 140155
- Mathmärchen*** 142157
- The binomial distribution 145160
- Wormhole points 147162
- You can't always get what you want 150165
- Matt Hu, Graduate student 152167
- Seeking successes 155170
- TEXNOΠAIΓNIA* 157172
- An area computation* 158173
- Experimental mathematics 161176
- Telescopes 163178
- The Rechner 166181
- Lipogram 168183
- 18th century machinations 170185
- Recipe* 173188
- Math talk* 175190
- Averages and estimates 178193
- Equality*** 180195
- pi, by parts* 183198
- A macaronic sonnet* 186201
- A hidden integral* 187202
- Letter to a princess* 189204
- Explicit Eulerian numbers 192207
- Mutperation satticsits*** 195210
- Review* 197212
- Plus C* 200215
- At the carnival** 203218
- Indicators** 206221
- Tweets** 209224
- Beautiful numbers** 210225
- Reminiscences*** 212227
- Winter journal** 215230
- Cellular automata 218233
- Tiling 221236
- Lattice points* 224239
- Afterlife* 227242
- Matt Hu and the Euler caper*** 230245
- Hypergeometric musings*** 233248
- On the bus*** 236251
- Style notes 239254
- Index 271286