# comb - FROM VOLUME 36 No 1 of The Book Review Column1 by...

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FROM VOLUME 36, No 1 of The Book Review Column 1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] A review of a book on Combinatorial Identities: Review 2 of Proofs that Really Count: The Art of Combinatorial Proof Author: Arthur T. Benjamin and Jennifer J. Quinn Publisher: MAA, 2003 \$43.95, Hardcover Reviewer: William Gasarch Abbott has been teaching Costello combinatorics. Abbott: Costello, how many subsets are there of { 1 , . . . , n } ? Costello: Oh. You can either choose 0 elements, or choose 1 element, or choose 2 elements, etc. So the answer is n i =0 ( n i ) . Abbott: Well . . . let me show you a different way to do it. The number 1 is either in the set A or not, so thats 2 choices. Then the number 2 is either in the set A or not, so thats 2 choices, etc. So the final answer is 2 × · · · × 2 = 2 n . So, Costello, you did the problem your way, I did it my way, and we got different answers. What can you conclude? Costello: That one of us is wrong? Abbott: No. We’ve shown. n i =0 ( n i ) = 2 n . Costello: Really! I don’t believe that! Prove it!! Abbott: We did! Costello: When? Abbott: Just now. Costello: What!? Abbott: Whats on Second. Costello: Who? Abbott: Who’s on first. Costello: (Ignoring reference) Usually when I do a math problem two ways and get two answers I assume one of them is wrong and try to find my error. Its better than what a friend of mine did in elementary algebra— do a problem three times and then take the average. Abbott: In math you can sometimes prove that two things are the same by solving the same problem two different ways. Costello: No way! Abbott: Way! Costello: I’d like to read more about this. Do you have a book to recommend? 1 c William Gasarch, 2005. 2 c William Gasarch, 2005 1

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Abbott: Yes. You should read Proofs that Really Count: The Art of Combinatorial Proof by Arthur T. Benjamin and Jennifer J. Quinn.
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