Exercises in (Mathematical) Style: Stories of Binomial Coefficients by John McCleary: MAA PRESS, 2017, XIV+275 PP., $48.00, ISBN: 978-1-4704-4783-0.

One is the well-known Vandermonde identity HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close=")" open="("><mfrac linethickness="0pt"><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mi>k</mi></mfrac></mfenced><mo>=</mo><msubsup><mo> </mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>k</mi></msubsup><mfenced close=")" open="("><mfrac linethickness="0pt"><mi>n</mi><mi>i</mi></mfrac></mfenced><mfenced close=")" open="("><mfrac linethickness="0pt"><mi>m</mi><mrow><mi>k</mi><mo>-</mo><mi>i</mi></mrow></mfrac></mfenced></mrow></math> ht . At this point, this very clever student invokes Newton's general binomial theorem to produce yet another power series from which emerges, but only after another page of ingenious simplifications, the elegant formula the professor was seeking: HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>C</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mfenced close=")" open="("><mfrac linethickness="0pt"><mrow><mn>2</mn><mi>n</mi></mrow><mi>n</mi></mfrac></mfenced></mrow></math> ht . However, after a few hours (and a half page of calculations), she managed to reduce this sum into a standard form for a general hypergeometric series involving a term HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow /><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></math> ht . [Extracted from the article]