Is mathematics discovered or

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1 996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies are groupigs of billios of stars boud together by gravity. Some galaxies, such as the Cetaurus galaxy show here, are elliptical i shape. Is mathematics discovered or iveted? For example, plaets revolve i elliptical orbits. Does that mea that the ellipse is out there, waitig for the mid to discover it? Or do people create the defiitio of a ellipse just as they compose a sog? Ad is it possible for the same mathematics to be discovered/iveted by idepedet researchers separated by time, place, ad culture? This is precisely what occurred whe mathematicias attempted to fid efficiet methods for raisig biomials to higher ad higher powers, such as 1x + 22, 1x + 22, 1x + 22, 1x , ad so o. I this sectio, we study higher powers of biomials ad a method first discovered/iveted by great mids i Easter ad Wester cultures workig idepedetly. Evaluate a biomial coefficiet. Biomial Coefficiets Before turig to powers of biomials, we itroduce a special otatio that uses factorials. Defiitio of a Biomial Coefficiet r For oegative itegers ad r, with Ú r, the expressio (read above r ) is called a biomial coefficiet ad is defied by r a r b =! r!1 - r2!. Techology Graphig utilities ca compute biomial coefficiets. For example, to fid 6, may utilities require the sequece 6 C r 2 ENTER. The graphig utility will display 1. Cosult your maual ad verify the other evaluatios i Example 1. The symbol C is ofte used i place of to deote biomial coefficiets. r r EXAMPLE 1 Evaluatig Biomial Coefficiets Evaluate: a. 6 b. c. 9 d. Solutio I each case, we apply the defiitio of the biomial coefficiet. a. a 6 2 b = 6! 2!16-22! = 6! 2!! = 6 # #! 2 # 1 #! = 1.

2 Sectio 1. The Biomial Theorem 997 b.!! 1 = = = =1!(-)!!! 1 c. Remember that! = 1. a 9 b = 9!!19-2! = 9!!6! = 9 # 8 # 7 # 6! # 2 # 1 # 6! = 8 d. a b =!!1-2! =!!! = 1 1 = 1 Check Poit 1 Evaluate: a. 6 b. c. 8 d. 6. Expad a biomial raised to a power. The Biomial Theorem Whe we write out the biomial expressio 1a + b2, where is a positive iteger, a umber of patters begi to appear. 1a + b2 1 = a + b 1a + b2 2 = a 2 + 2ab + b 2 1a + b2 = a + a 2 b + ab 2 + b 1a + b2 = a + a b + 6a 2 b 2 + ab + b 1a + b2 = a + a b + 1a b 2 + 1a 2 b + ab + b Each expaded form of the biomial expressio is a polyomial. Observe the followig patters: 1. The first term i the expasio of 1a + b2 is a. The expoets o a decrease by 1 i each successive term. 2. The expoets o b i the expasio of 1a + b2 icrease by 1 i each successive term. I the first term, the expoet o b is. (Because b = 1, b is ot show i the first term.) The last term is b.. The sum of the expoets o the variables i ay term i the expasio of 1a + b2 is equal to.. The umber of terms i the polyomial expasio is oe greater tha the power of the biomial,. There are + 1 terms i the expaded form of 1a + b2. Usig these observatios, the variable parts of the expasio of 1a + b2 6 are a 6, a b, a b 2, a b, a 2 b, ab, b 6. The first term is a 6, with the expoets o a decreasig by 1 i each successive term. The expoets o b icrease from to 6, with the last term beig b 6. The sum of the expoets i each term is equal to 6. We ca geeralize from these observatios to obtai the variable parts of the expasio of 1a + b2. They are a, a 1 b, a 2 b 2, a b,..., ab 1, b. Expoets o a are decreasig by 1. Expoets o b are icreasig by 1. Sum of expoets: = Sum of expoets: + = Sum of expoets: =

3 998 Chapter 1 Sequeces, Iductio, ad Probability If we use biomial coefficiets ad the patter for the variable part of each term, a formula called the Biomial Theorem ca be used to expad ay positive itegral power of a biomial. A Formula for Expadig Biomials: The Biomial Theorem For ay positive iteger, 1a + b2 = a + 1 a - 1 b + a - 2 b 2 + a - b + Á + b = a r = r a - r b r. EXAMPLE 2 Usig the Biomial Theorem Expad: 1x Solutio We use the Biomial Theorem 1a + b2 = a + 1 a - 1 b + a - 2 b 2 + a - b + Á + b to expad 1x I 1x + 22, a = x, b = 2, ad =. I the expasio, powers of x are i descedig order, startig with x. Powers of 2 are i ascedig order, startig with 2. (Because 2 = 1, a 2 is ot show i the first term.) The sum of the expoets o x ad 2 i each term is equal to, the expoet i the expressio 1x Techology You ca use a graphig utility s table feature to fid the five biomial coefficiets i Example 2. Eter y 1 = Cr x. AB A2 B A B A1 B A B (x+2) =a bx +a bx 1 2+a bx a bx 2 +a b2! These biomial coefficiets are evaluated usig Ar B = r!( r)!. =!!! x +! 1!! x 2+! 2!2! x2 +!! x 8+ 16!1!!!! 2!2! = 2! 12 2! 2 1 = 2 = 6 Take a few miutes to verify the other factorial evaluatios. =1 x +x 2+6x 2 +x = x + 8x + 2x 2 + 2x + 16 Check Poit 2 Expad: 1x EXAMPLE Usig the Biomial Theorem Expad: 12x - y2. Solutio Because the Biomial Theorem ivolves the additio of two terms raised to a power, we rewrite 12x - y2 as 2x + 1-y2. We use the Biomial Theorem 1a + b2 = a + 1 a - 1 b + a - 2 b 2 + a - b + Á + b

4 Sectio 1. The Biomial Theorem 999 to expad 2x + 1-y2. I 2x + 1-y2, a = 2x, b = -y, ad =. I the expasio, powers of 2x are i descedig order, startig with 12x2. Powers of -y are i ascedig order, startig with 1-y2. [Because 1-y2 = 1, a -y is ot show i the first term.] The sum of the expoets o 2x ad -y i each term is equal to, the expoet i the expressio 12x - y2. 12x - y2 = 2x + 1-y2 =a b(2x) +a b(2x) 1 ( y)+a b(2x) 2 ( y) 2 +a b(2x) 2 ( y) +a b(2x)( y) +a b( y)! Evaluate biomial coefficiets usig Ar B = r!( r)!.!!!! (2x) + (2x)! ( y)+ 1!! 2!! (2x) ( y) 2! + (2x) 2 ( y)! + (2x)( y)! = + ( y)!2!!1!!!!! 2!! = 2 1! = 1 Take a few miutes to verify the other factorial evaluatios. =1(2x) +(2x) ( y)+1(2x) ( y) 2 +1(2x) 2 ( y) +(2x)( y) +1( y) Raise both factors i these paretheses to the idicated powers. =1(2x )+(16x )( y)+1(8x )( y) 2 +1(x 2 )( y) +(2x)( y) +1( y) Now raise y to the idicated powers. = 112x x 21-y x 2 y x 2 21-y x2y + 11-y 2 Multiplyig factors i each of the six terms gives us the desired expasio: 12x - y2 = 2x - 8x y + 8x y 2 - x 2 y + 1xy - y. Check Poit Expad: 1x - 2y2. Fid a particular term i a biomial expasio. Fidig a Particular Term i a Biomial Expasio By observig the terms i the formula for expadig biomials, we ca fid a formula for fidig a particular term without writig the etire expasio. 1st term 2d term rd term a bab a 1 ba 1b1 a 2 ba 2b2 The expoet o b is 1 less tha the term umber. Based o the observatio i the bottom voice balloo, the 1r + 12st term of the expasio of 1a + b2 is the term that cotais b r. Fidig a Particular Term i a Biomial Expasio The 1r + 12st term of the expasio of 1a + b2 is r a - r b r.

5 1 Chapter 1 Sequeces, Iductio, ad Probability EXAMPLE Fidig a Sigle Term of a Biomial Expasio Fid the fourth term i the expasio of 1x + 2y2 7. Solutio The fourth term i the expasio of 1x + 2y2 7 cotais 12y2. To fid the fourth term, first ote that = + 1. Equivaletly, the fourth term of 1x + 2y2 7 is the st term. Thus, r =, a = x, b = 2y, ad = 7. The fourth term is a 7 7 b(x) 7 (2y) =a b(x) (2y) = 7!!(7-)! (x) (2y). Use the formula for the (r + 1)st term of (a + b) : Ar Ba r b r.! We use Ar B = r!( r)! 7 to evaluate AB. Now we eed to evaluate the factorial expressio ad raise x ad 2y to the idicated powers. We obtai 7!!! 181x 218y 2 = 7 # 6 # #! # 2 # 1 # 181x 218y 2 = 181x 218y 2 = 22,68x y.! The fourth term of 1x + 2y2 7 is 22,68x y. Check Poit Fid the fifth term i the expasio of 12x + y2 9. The Uiversality of Mathematics Pascal s triagle is a array of umbers showig coefficiets of the terms i the expasios of 1a + b2. Although credited to Frech mathematicia Blaise Pascal ( ), the triagular array of umbers appeared i a Chiese documet prited i 1. The Biomial Theorem was kow i Easter cultures prior to its discovery i Europe. The same mathematics is ofte discovered/iveted by idepedet researchers separated by time, place, ad culture. Biomial Expasios Pascal s Triagle Chiese Documet: 1 Coefficiets i the Expasios 1a + b2 = 1 1 1a + b2 1 = a + b 1 1 1a + b2 2 = a 2 + 2ab + b a + b2 = a + a 2 b + ab 2 + b 1 1 1a + b2 = a + a b + 6a 2 b 2 + ab + b a + b2 = a + a b + 1a b 2 + 1a 2 b + ab + b

6 Sectio 1. The Biomial Theorem 11 Exercise Set 1. Practice Exercises I Exercises 1 8, evaluate the give biomial coefficiet A - x - x B 2. 2 x x f1x + h2 - f1x2 I Exercises, fid h ad simplify.. f1x2 = x + 7. f1x2 = x I Exercises 9, use the Biomial Theorem to expad each biomial ad express the result i simplified form. 9. 1x x x + y x + y2 1. 1x x x x x 2 + 2y x + y y y x x c c x x x - y x - y a + b2 6. 1a + 2b2 6 I Exercises 1 8, write the first three terms i each biomial expasio, expressig the result i simplified form. 1. 1x x x - 2y x - 2y2. 1x x y y I Exercises 9 8, fid the term idicated i each expasio x + y2 6 ; third term. 1x + 2y2 6 ; third term 1. 1x ; fifth term 2. 1x ; fifth term. 1x 2 + y 2 8 ; sixth term. 1x + y ; sixth term. Ax - 1 fourth term 6. Ax B 9 ; 2 B 8 ; fourth term 7. 1x 2 + y2 22 ; the term cotaiig 8. 1x + 2y2 1 ; the term cotaiig y 1 y 6. Fid the middle term i the expasio of a x + x 1. b 6. Fid the middle term i the expasio of Applicatio Exercises a 1 12 x - x2 b. The graph shows that U.S. smokers have a greater probability of sufferig from some ailmets tha the geeral adult populatio. Exercises 7 8 are based o some of the probabilities, expressed as decimals, show to the right of the bars. I each exercise, use a calculator to determie the probability, correct to four decimal places. Probability That Uited States Adults Suffer from Various Ailmets Tobacco-Depedet Populatio Depressio Frequet Hagovers Axiety/Paic Disorder Severe Pai Source: MARS 2 OTC/DTC Geeral Populatio Probability If the probability a evet will occur is p ad the probability it will ot occur is q, the each term i the expasio of 1p + q2 represets a probability. 7. The probability that a smoker suffers from depressio is.28. If five smokers are radomly selected, the probability that three of them will suffer from depressio is the third term of the biomial expasio of ( ). smokers are selected. Practice Plus I Exercises 9 2, use the Biomial Theorem to expad each expressio ad write the result i simplified form. 9. 1x + x x 2 + x - 2 Probability a smoker suffers from depressio What is this probability? Probability a smoker does ot suffer from depressio

7 12 Chapter 1 Sequeces, Iductio, ad Probability 8. The probability that a perso i the geeral populatio suffers from depressio is.12. If five people from the geeral populatio are radomly selected, the probability that three of them will suffer from depressio is the third term of the biomial expasio of What is this probability? Writig i Mathematics 9. Explai how to evaluate Provide a example with your r. explaatio. 6. Describe the patter o the expoets o a i the expasio of 1a + b Describe the patter o the expoets o b i the expasio of 1a + b What is true about the sum of the expoets o a ad b i ay term i the expasio of 1a + b2? 6. How do you determie how may terms there are i a biomial expasio? 6. Explai how to use the Biomial Theorem to expad a biomial. Provide a example with your explaatio. 6. Explai how to fid a particular term i a biomial expasio without havig to write out the etire expasio. 66. Describe how you would use mathematical iductio to prove What happes whe = 1? Write the statemet that we assume to be true. Write the statemet that we must prove. What must be doe to the left side of the assumed statemet to make it look like the left side of the statemet that must be proved? (More detail o the actual proof is foud i Exercise 8.) Techology Exercises C r 67. Use the key o a graphig utility to verify your aswers i Exercises 1 8. I Exercises 68 69, graph each of the fuctios i the same viewig rectagle. Describe how the graphs illustrate the Biomial Theorem. 68. Probability a perso i the geeral populatio suffers from depressio ( ). Probability a perso i the geeral populatio does ot suffer from depressio 1a + b2 = a + 1 a - 1 b + a - 2 b 2 + Á ab b. people from the geeral populatio are selected. f 1 1x2 = 1x + 22 f 2 1x2 = x f 1x2 = x + 6x 2 f 1x2 = x + 6x x f 1x2 = x + 6x x + 8 Use a -1, 1, 1 by -,, 1 viewig rectagle. 69. f 1 1x2 = 1x + 12 f 2 1x2 = x Use a -,, 1 by -,, 1 viewig rectagle. I Exercises 7 72, use the Biomial Theorem to fid a polyomial expasio for each fuctio. The use a graphig utility ad a approach similar tothe oe i Exercises 68 ad 69 to verify the expasio. 7. f 71. f 1 1x2 = 1x x2 = 1x f 1x2 = x + x f 1x2 = x + x + 6x 2 f 1x2 = x + x + 6x 2 + x f 6 1x2 = x + x + 6x 2 + x + 1 f 1 1x2 = 1x Critical Thikig Exercises Make Sese? I Exercises 7 76, determie whether each statemet makes sese or does ot make sese, ad explai your reasoig. 7. I order to expad 1x - y 2, I fid it helpful to rewrite the expressio iside the paretheses as x + 1-y Without writig the expasio of 1x , I ca see that the terms have alteratig positive ad egative sigs. 7. I use biomial coefficiets to expad 1a + where b2, is 1 the coefficiet of the first term, is the coefficiet of the secod term, ad so o. 76. Oe of the terms i my biomial expasio is 7 x2 y. I Exercises 77 8, determie whether each statemet is true or false. If the statemet is false, make the ecessary chage(s) to produce a true statemet. 77. The biomial expasio for 1a + b2 cotais terms. 78. The Biomial Theorem ca be writte i codesed form as 79. The sum of the biomial coefficiets i 1a + b2 caot be There are o values of a ad b such that 81. Use the Biomial Theorem to expad ad the simplify the result: 1x 2 + x Hit: Write x 2 + x + 1 as x 2 + 1x Fid the term i the expasio of 1x 2 + y 2 2 cotaiig x as a factor. 8. Prove that 8. Show that Hits: 1a + b2 = a r = 1a + b2 = a + b. r = - r. r + r + 1 = + 1 r r2! = 1 - r21 - r - 12! 1r + 12! = 1r + 12r! r a - r b r.

8 Sectio 1.6 Coutig Priciples, Permutatios, ad Combiatios 1 8. Follow the outlie below ad use mathematical iductio to prove the Biomial Theorem: 1a + b2 = a + 1 a - 1 b + a - 2 b 2 + Á ab b. a. Verify the formula for = 1. b. Replace with k ad write the statemet that is assumed true. Replace with k + 1 ad write the statemet that must be proved. c. Multiply both sides of the statemet assumed to be true by a + b. Add expoets o the left. O the right, distribute a ad b, respectively. d. Collect like terms o the right. At this poit, you should have 1a + b2 k + 1 = k ak B k + k 1 Rak b + B k 1 + k Rak - 1 b 2 + B k + k Rak - 2 b f. Because k (why?) ad = k + 1 (why?), substitute these results ad the results from part (e) ito the equatio i part (d). This should give the statemet that we were required to prove i the secod step of the mathematical iductio process. Preview Exercises Exercises will help you prepare for the material covered i the ext sectio.! 86. Evaluate for = 2 ad r =. 1 - r2!! 87. Evaluate for = 8 ad r =. 1 - r2! r! k k = k + 1 k You ca choose from two pairs of jeas (oe blue, oe black) ad three T-shirts (oe beige, oe yellow, ad oe blue), as show i the diagram. + Á k + B k k k Rabk + k k bk + 1. e. Use the result of Exercise 8 to add the biomial sums i brackets. For example, because r + r + 1 = + 1 r + 1, the k + k 1 = k + 1 ad 1 k 1 + k = k True or false: The diagram shows that you ca form 2 *, or 6, differet outfits. Sectio Objectives Use the Fudametal Coutig Priciple. Use the permutatios formula. Distiguish betwee permutatio problems ad combiatio problems. Use the combiatios formula. 1.6 Coutig Priciples, Permutatios, ad Combiatios Have you ever imagied what your life would be like if you wo the lottery? What chages would you make? Before you fatasize about becomig a perso of leisure with a staff of obediet elves, thik about this: The probability of wiig top prize i the lottery is about the same as the probability of beig struck by lightig. There are millios of possible umber combiatios i lottery games ad oly oe way of wiig the grad prize. Determiig the probability of wiig ivolves calculatig the chace of gettig the wiig combiatio from all possible outcomes. I this sectio, we begi preparig for the surprisig world of probability by lookig at methods for coutig possible outcomes.