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... T O P I C S IN A L G E B R A 8y HAROLD J. GAY and RAYMOND K. MORLEY, o. Ph. Professors of ·Mathematics Worcester Polytechnic Institute Worcester, Massachusetts 1942 CopJrllht 1931 lMl 1 J. BAROLD GAY RAYMOND K. IIORLEY _..tett m u.a..L IJ&. . BDWAilDS BROTHERS, INC. AHN Aaaoa. MICHIGAN 19fR PO REWORD . • This book has been prepared b7 the authors to meet the requirements of the Preshman Mathematics courses given at the Worcester Pol7technic Institute. It is their opinion tbat this selection of topics rrom the field' of College Algebra those that are comp~ises most essential 1n the training of students ror the en gineering • pro~eas1on • Within the book itself, Morle7 1s re- ~cto~ sponsible. the preparation of the material in Chap fa~ ter IV, and Gay for that in Chapters I, II, Professo~ III. and The authors v1sh to express their thanks to their colleagues, Professor B&rr1·a Rice, Pro~essor William L. Phinney, Jr., •dward C. Brown, Professo~ and Mr. David for their generous assistance 1n ~leT checking the text .and the answers. Harold J. Ga7 Morle7 Ray.mond K •. Worcester. Massachusetts November; 19'9 ... CONTENTS Chapter Page I. Combinations, and Prob- Pe~utat1on~, ab111 t7. • • • • . • • • • • • • • • • • • 1 ~ . . II. Determinants • • • • ' • • • • • • • • 45 Imaginary Nuiilbers • .I~I. • • • • • • • • • • • 61 IV·.. Solution of Equations • • • • • • • • • I . . . . . . . . . ·- . . . . . . Answers . 87 ' ... TOPICS IN ALGEBRA Chapter I PERMD'rATION5, COMBINATIONS, AND PROBABILI'l'Y I-1. INTRODUCTION. The meaning of permutat1on and 11 11 "combination" may be explained best considering them simul by taneously. Suppose we have a group of objects; to be concrete, five flags of different colors, red, blue, green, yellow, and black. If ve select three of them, say the red, blue, and green, . . paying no attention to the order of select1ons or of their ar~ 1 rangement with respect to each in any way, it is said that othe~ ve have taken a "combination of the :five flags three at a time." Of course, there are many possible combinations of the othe~ . five flags three at a time. · Next suppos·e that we take three of the five flags and them in a rov tor a signal, each different ot the ~1splay o~der flags 1n the row being a signal according to a certain d1ffere~t code. Bach such possible order ot the flags is called a "permu tation of the five flags three at a time." These examples are in accord with the following def1n1- tiona. Def1n1t1on. An arrangement of given number of elements ~ is called permutation. ~ Definition. Anl one of the sets into which number of ~ elements m&l be no aocount beias taken of the possible srou~ed, ot the set, is called .! comb·ination. E.~rmutat1ons In problems 1nvolv1rig per.mutationa and combinations the objeot is to find the total·number possible. In the following sections we shall discuss formulas and methods for doing this. The atudT of is closely allied to per.muta p~obab111~y t1ons and combinations; many problems in probability are based on them ao 1t is to group all in one chapter approp~1ate th~ee leaving the d1scusa1on of probability the last. fo~ I. PERMUTATIONS - I-2. PBRMUTA!'IONS OJP -n BI·EME:NTS TAXD -r AT A TIME. We state the following Fundamental Principle. If' an act which can be pel!_~ormed in &nl ot a va7a is an act which can be performed in ~allowed ~ aqz ot b wars, the number of wars in which the tvo acts !B;l be performed in succession is ab. the.produc~ 11 PBRMUTATIONS, COMBINATIONS, AID PROBABILI'l'Y 20. How many different hands of 5 cards ma7 be dealt from a pack, eaoh.hand containing 3 aces and 2 other cards of the same rank ( kind ) ? 21. How many hands of 5 cards may be dealt from di~te~ent a pack (a) all of the heart suit? (b) All of the same suit? . 5 22. Uow many different hands of cards may be dealt from a pack all o£ the suit and in sequence? s~e 23. From a -pack of cards in how many ways may 7 cards of the same suit be arranged in a on a table? ~ow • 24 • Fz-om a pack of cards 1n how many ways may 4 hands o:r 1' cards each be dealt and distributed to 4 pla7ers? (Leave· in factored for.m.) answe~ 25. From a group of 12 1n how many ways may (a) a ma~1nes or squad of 8 be chosen? (b) A squad 8 be chosen and arranged in two ranks of men each? ~ 26. A has 10 friends, among them X and Z who are not man on speaking tezam.s. In· how lD8Jll'" ways may he invite 6 of them to .a part7 it both X and Z-·ma7 not be included? 5 27. There are candidates a society. In how many fo~ of ways may groups of one or them be summoned the mc~e befo~e societ-y? or 28. How man7 different groups one or may mo~e acto~s appear on the stage at once if there are 10 actors in the cast? or 29. An examination paper. contains 10 questions which nos. 1, 2, and 3 must be answered. In how many ways may a stu dent choose his questions if he must answeP (a) just 7 questions? (b) At least 7 ques.tions? ·. ,a. In a twilight· baseball league there are 10 clubs each at which pla7s 7 games with every other clu)l. Bov many games must be scheduled for the whole league? 31. If a state makes automobile registration plates which contain just two letters of the alphabet, excluding I and 0 ~al 1 lowed by numbers of one., two, or three digits, of which the .. r1rst digit may not be zero,. how many automobiles may be ~egis tared? III . PROBABILITY I-6. PROBABILITY. ·The words "pl'oba.ble and "probabili 11 ty•• are used :rather loosely in evei7d&7 life. In what follows we shall consider first mathematical probab111tZJ and second, .e][Perimental, c,r emp1l'ioal, p~obab111tr. - It s denotes the number of va7s in which it is equally . ! likel7 that an event ma7 ocour, denotes the number ot and Y&7&