combinatorics 2nd ed. - r. merris

[...]... complete without some errors An up-to-date errata, accessible from the Internet, will be maintained at URL http://www.sci.csuhayward.edu/$rmerris Appropriate acknowledgment will be extended to the first person who communicates the specifics of a previously unlisted error to the author, preferably by e-mail addressed to merris@ csuhayward.edu Hayward California RUSSELL MERRIS 1 The Mathematics of Choice... 20,736 different 4-letter ‘‘words’’ could be constructed using the 12-letter Hawaiian alphabet 1.1 Exercises 7 (b) Show that 456,976 different 4-letter ‘‘words’’ could be produced using the 26-letter English alphabet.* (c) How many four-letter ‘‘words’’ can be assembled using the Hawaiian alphabet if the second and last letters are vowels and the other 2 are consonants? (d) How many four-letter ‘‘words’’... three-number combination, how long would it take to try all possible combinations? 20 The International Standard Book Number (ISBN) is a 10-digit numerical code for identifying books The groupings of the digits (by means of hyphens) varies from one book to another The first grouping indicates where the book was published In ISBN 0-8 817 5-0 8 3-2 , the zero shows that the book was published in the English-speaking... L ¼ 11 À R (When R ¼ 1, the check digit is X.) (c) What is the value of the check digit, L, in ISBN 0-5 3 4-9 5154-L? (d) Unlike POSTNET, the more sophisticated ISBN system can not only detect common errors, it can sometimes ‘‘correct’’ them Suppose, e.g., that a single digit is wrong in ISBN 9 0-5 59 9-0 7 8-0 Assuming the check digit is correct, can you identify the position of the erroneous digit? (e) Now... x3 g corresponds to the nine-leter word YYYNNNNNN In general, there is a one-to-one correspondence between subsets of fx1 ; x2 ; ; xn g, and n-letter words assembled from the alphabet fN; Yg Moreover, in this correspondence, r-element subsets correspond to words with r Y’s and n À r N’s We seem to have discovered a new way to think about Cðn; rÞ It is the number of n-letter words that can be produced... Section 1.7, one is led to symmetric polynomials and, in Section 1.8, to partitions of n Elementary symmetric functions and their association with power sums lie at the Combinatorics, Second Edition, by Russell Merris ISBN 0-4 7 1-2 6296-X # 2003 John Wiley & Sons, Inc 1 2 The Mathematics of Choice heart of Section 1.9 The final section of the chapter is an optional introduction to algorithms, the flavor... different ways can an r-element subset be chosen from an n-element set S? Denote the number by Cðn; rÞ Pronounced ‘‘n-choose-r’’, Cðn; rÞ is just a name for the answer Let’s find the number represented by this name Some facts about Cðn; rÞ are clear right away, e.g., the nature of the elements of S is immaterial All that matters is that there are n of them Because the only way to choose an n-element subset... published in the English-speaking world The code for the Netherlands is ‘‘90’’ as, e.g., in ISBN 9 0-5 69 9-0 7 8-0 Like POSTNET, ISBN employs a check digit scheme The first nine digits (ignoring hyphens) are multiplied, respectively, by 10, 9, 8; ; 2, and the resulting products summed to obtain S In 0-8 817508 3-2 , e.g., S ¼ 10 Â 0 þ 9 Â 8 þ 8 Â 8 þ 7 Â 1 þ 6 Â 7 þ 5 Â 5 þ 4 Â 0 þ 3 Â 8 þ 2 Â 3 ¼ 240: The... next-to-last touch is at the point ðr; rÞ) (c) cn is the nth Catalan number of Exercises 13–14, n ! 1 17 Let X and Y be disjoint sets containing n and m elements, respectively In how many different ways can an ðr þ sÞ-element subset Z be chosen from X [ Y if r of its elements must come from X and s of them from Y? 18 Packing for a vacation, a young man decides to take 3 long-sleeve shirts, 4 short-sleeve... Proof Consider the ðn þ 1Þ-element set fx1 ; x2 ; ; xn ; yg Its r-element subsets can be partitioned into two families, those that contain y and those that do not To count the subsets that contain y, simply observe that the remaining r À 1 elements can be chosen from fx1 ; x2 ; ; xn g in Cðn; r À 1Þ ways The r-element subsets that do not contain y are precisely the r-element subsets of & fx1 . person who communi- cates the specifics of a previously unlisted error to the author, preferably by e-mail addressed to merris@ csuhayward.edu R USSELL MERRISHayward California Preface xi 1 The. some errors. An up-to-date errata, accessible from the Internet, will be maintained at URL http://www.sci.csuhayward.edu/$rmerris Appropriate acknowledgment will be extended to the first person. correspondents Mark Hunacek and Gerhard Ringel; reviewers Rob Beezer, John Emert, Myron Hood, Herbert Kasube, Andre ´ Ke ´ zdy, Charles Landraitis, John Lawlor, and Wiley editors Heather Bergman,