SAMPLE College Math
THE ULTIMATE CREDIT-BY-EXAM STUDY GUIDE FOR: College Math 2 nd Edition
12/01/2023
Acknowledgements Wewouldliketothanktheauthorfortheirpatience,support,andexpertiseincontributingtothis studyguide;andoureditorsfortheirinvaluableeffortsinreadingandeditingthetext.Wewould also like to thank those at Achieve Test Prep whose hard work and dedication to fulfilling this project did not go unnoticed.Lastly,wewouldliketothanktheAchieveTestPrepstudentswho have contributed to the growth of these materials over the years.
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Contents
Chapter 1: Introduction to Algebra and Functions
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A. Set of Real Numbers
1 3 4 5 5 6 7 7
B. Inequalities
C. Mathematical Translations of Equalities from Words D. Mathematical Translations of Inequalities from Words
E. Solving Equations F. Solving Inequalities
G. Solving Inequalities with Modulus Symbol
H. Systems of Linear Equations
I. Polynomials
12 13 14 16 17 19 22 26 27 28 29 29 29 30 31 32 32 33 34 36 38 41 42 44 45 26 38
J. Solving Quadratic Equations
K. Functions
L. Exponential Functions M. Logarithm Functions Chapter 1: Review Questions Chapter 1: Review Answers
Chapter 2: Financial Mathematics
A. Percent
B. Discounts, Markups, and Taxation
C. Change in Percentage D. Profits and Losses
E. Interest
F. Simple Interest
G. Compound Interest H. Continuous Interest
I. Effective Interest Rate (Effective Annual Rate of Interest or EAR)
J. Effective Annual Yield/Annual Percentage Rate (APR)
K. Present and Future Value Chapter 2: Review Questions Chapter 2: Review Answers
Chapter 3: Geometry
A. Triangles
B. Quadrilaterals
C. Parallel and Perpendicular Lines
D. Circles
E. Theorems of Circle
Chapter 3: Review Questions Chapter 3: Review Answers
47 49
Chapter 4: Counting and Probability
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A. Counting
53 54 55 56 56 57 58 59 60 61 61 63 66 70 70 72 74 74 75 76 77 77 78 79 82 84 85 85 85 85 86 87 70 84
B. Fundamental Counting Principle
C. Permutations
D. Distinguishable Permutations
E. Combinations
F. Probability
G. Complement of an Event H. Mutually Exclusive Events
I. Intersection of Independent Events
J. Expected Value
K. Conditional Probability Chapter 4: Review Questions Chapter 4: Review Answers
Chapter 5: Data Analysis and Statistics
A. Data Interpretation
B. Data Tables C. Bar Graph D. Line Graphs E. Pie Charts
F. Measures of Central Tendency and Measures of Dispersion
G. Mean
H. Median
I. Mode
J. Standard Deviation and Variance Chapter 5: Review Questions
Chapter 5: Review Answers
Chapter 6: Logic and Sets
A. Logical Operations and Statements
B. Negation
C. Conjunction D. Disjunction E. Parentheses
F. Truth Values and Truth Tables
G. Conditional Statements
H. Logical Equivalence
87 87 88 91 92 95 96 99
I. Converse, Inverse, and Contrapositive
J. Logical Arguments
K. Sets
L. Operations on Sets M. Cartesian Products
Chapter 6: Review Questions Chapter 6: Review Answers
Chapter 1: Introduction to Algebra and Functions
Overview Mathematics isacommontoolthatisusedineverydaylife.Rangingfromsimplecountingofinventory items to solving complex equations in computer and engineering work, every day involves the useof mathematics.Therearealargenumberofincentivesforanindividualtostudythroughoutthiscourseto equip him/herself with the knowledge of mathematics in order to excel in their career. Mathematics providesuswithabetterunderstandingoftheworldaroundus.Ithelpstohonetheindividual’sskillsin problem solving, logical reasoning and flexible thinking, which is ofutmostimportanceinthemodern businessworld.Itispervasiveandusefulinalmostallthearenasoflife,suchasbusinessmanagement, predicting stock market prices, safeguarding credit transactions on the internet,science,engineering, and even managing day-to-day financial activities. In this chapter, the introduction to algebraandfunctionswillbepresented.Themainfocuswillbeon concept building and its application.Thebuildingblocksofthischapterwillincludesolvingequations, linearinequalities,systemsoflinearequationsbyanalyticalandgraphicalmethods,functions,andlinear and exponential growth.Theaimofthischapteristoequipyouwiththeknowledgeofgeneralalgebra that helps in building the foundation of mathematics. Objectives A. Set of Real Numbers Algebraisatoolthatisusedtosolvereallifeproblemsinthedomainsofscience,business,architecture, management, space travel, and many other fields. We begin this chapter by understanding the basic notations and symbols used to build and solve algebraic expressions. Sets represent the collection of similar elements, and are denoted by the enclosed brackets {}. The uniquecharacteristicofallelementsinasetisthattheyhavesomesimilarityinappearanceofpurpose. Forinstance,{2,4,6,8}representsthesetofallone-digitevennumbers,and{a,e,i,o,u}representsthe setofallvowelsinEnglish.Setsaredenotedbyaletter.Thesetofallpositivenumberslessthan10is denotedbyS={1,2,3,4,5,6,7,8,9}andthesetofalloddnumberslessthan10isdenotedbyT={1, 3,5,7,9}.Inthiscase,Tisknownasthesub-setofS,sinceallelementsinsetTarepresentinsetS. Another way to describe a set is by using a set-builder notation. For instance, the set {1, 3, 5} in a set-buildernotationiswrittenas{xlxisanoddnumberbetween0and6}.Set-buildernotationisread as follows: By the end of this chapter, you should be able to recognize, understand, and solve the following: ● Algebraic operations ● Equations and inequalities ● Functions and their properties ● Number systems and operations
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Real numbers are used extensively in mathematicsandarerepresentedonacontinuousnumberline. Thenumberlineincludesthesetofallthewholenumbers{0,1,2,3,4…},naturalnumbers{1,2,3,4…}, integers{…-3,-2,-1,0,1,2,3,…},fractions(numberswrittenintheformof ,suchthattheratioisnot equaltozero)andirrationalnumbers(likesquarerootof2,givenby ).Onthenumberline,negative 2 values are on the left side of zero while positive values are on the right side of zero, as shown below: The properties of real numbers deal with four major operations: addition (+), subtraction ( ), − multiplication ( ) and division ( ). Addition, subtraction, and multiplication of real numbers will ×, ∙ ÷, / result inarealnumber.Divisionofrealnumbersisalsopossible,providedthatthedenominatorofthe fraction is not equal to zero. If the denominator is equal to zero, then the result is undefined. The Properties of Real Number are listed below (assume , , and are real numbers): 1. Cumulative law states that when we add or multiply two real numbers, their order does not matter. That is, and + = + × = × 2. Associative law states that while adding or multiplying more than two real numbers thenthe effect of parentheses does not matter. That is, and ( + ) + = + ( + ) ( × )× = ×( × ) 3. Distributive law applies to the caseswhereonemakesuseofbothadditionandmultiplication operations. and ×( + ) = × + × ( + )× = × + × 4. The additive identity is known as 0 and 1 is a multiplicative identity. That is, and + 0 = ×1 = 5. Additiveinverseisdenotedbyanegativesign.Thatis,theadditiveinverseof wouldbe( ). − Addition of a real number with its additive inverse would be 0. So, + (− ) = 0 6. Multiplicative inverse of would be -1 or . The multiplication of a real number with its −1 1 multiplicative inverse would be equal to 1, Therefore, × −1 = 1 7. Cancellation law for addition: if , then . + = + = 8. Cancellation law for multiplication: if , then , such that . × = × = ≠0 9. Cancellation law for division: , provided and . = ≠0 ≠0 Absolute values, or the magnitude of an integer, are another vital concept to know in algebra. It is definedasthedistancebetweentheintegerandthezerovalueonthenumberline.Absolutevaluesare denoted by the modulus symbol where can be any integer, negative or positive. If we have to | |
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calculate the absolute value of -5, then it will be denoted by |-5|and is equal to 5, showing that the distance from -5 to 0 is 5 units. Some of the properties of absolute values are: | | = { ≥0 − ( ) < 0 || − | ≥ | 0= | | | * | = | | * | | Scientific Notation Sometimeswemakeuseofsomepowerof10thatmakesitconvenienttowriteaverylargenumber;this means we are making use of scientific notation. For instance, suppose we have to write the number 687657.788. We can write it as . 6. 87657788×10 5 B. Inequalities The value of different numbers can be compared by their relative position on the number line. For instance, in the given number line below,
<
islessthan andisdenotedby also say , that is isgreaterthan andliesontherightsideofthenumberline,relativetothe > position of . The mathematical representation for different expressions is given in the following table: ,whichmeans liestotheleftof onthenumberline.Wecan
= < > ≥ ≤ ≠ ≈
Interpretations is equal to is less than is greater than is greater than or equal to is less than or equal to is not equal to
Expressions
is approximately equal to The symbols <, >, ≥,≤and≠denotethesignsofinequalitiesandtheexpressionsusingthesesignsof inequalities, , , , and denotes inequalities. We can make use of set-builder < > ≥ ≤ ≠ notationfortheseinequalitiesaswell.Forinstance, meansset representsallthevalues = { | ≥2} of such that is greater than or equal to 2. This can be shown graphically on the number line as follows:
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It must benotedthatintheabovenumberline,thereisaclosedbracket[at2,whichmeansthat2is alsoincludedintheexpression.Incontrast,theexpression isrepresentedonthenumber = { | > 2} line as follows:
Whenaparenthesis(isusedonthenumberline,itmeansthatallnumbersgreaterthan2,excluding2, aretobeconsideredhere.Inotherwords,whileusinginequalities,wemaymakeuseofdifferenttypes of brackets that have different meanings. For instance, means the set contains allthenumbers ( , ) between and , excluding and . On the contrary, means all numbers between and , [ , ] including and . Similarly, meansallnumbersbetween and ,inclusiveof andexclusiveof .Thisisknownan [ , ) interval notation. Let us understand this using a few examples: Set-Builder Notation Graphical Representation Interval Notation { | > 4} (4, ∞) { | < 2} (− ∞, 2) { | ≤ 1 } (− ∞, 1] { | ≥ − 3} [− 3, ∞) C. Mathematical Translations of Equalities from Words Anequationisusedtoshowthattwomathematicalexpressionsareequaltoeachother. Forinstance, . This is as an equation since theexpressionsonbothsidesofthesentenceareequaland 2 + 5 = 9 denoted by the equal to (=) sign. Some of the examples of equations are: 3 + 5 = 8 2 5 7 0 * – 1 3 0 = = = 2 8 8 1 – 0 0 2 In these examples, the expressions on either side of the equal sign are equal to each other.
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Writtensentencescanbeconvertedintomathematicalexpressionssothattheycanbesolvedtoarrive at meaningful solutions. For instance, “a number is multiplied by 10 and then added to 8toget48”. Assume the number is and the mathematical equation becomes: 10 + 8 = 48 It shouldbenotedthatthetwosidesoftheequationshouldbalanceeachotherusingtheequalto(=) sign. Let us solve a few more examples to get a better understanding.
Word Problem
Algebraic Translation
( 2 2+ 1 0 ) = 35 2 – 6 – 5 = 3
Kathyaskshermotherhowoldsheis.Hermotherreplies,“Ifyou double my age and add 10 to it, then divide by 2, you get 35.”
Marlingetshismonthlypocketmoney.Hesaveshalfofit,andthen makes two purchases worth $6 and $5. Finally whenhereaches home, he is left with $3.
A bus starts leaves the station filledtocapacity.Atthefirststop, halfthepassengersgetoffand10getonboard.Atthenextstation, 12getoffwhile15getonboard.Now,thenumberofpassengersin the bus is 28.
− 2 + 10 − 12 + 15 = 28
D. Mathematical Translations of Inequalities from Words Inequalities can also be translated from wordsintomathematicalequations.Forinstance,weneedto translate the sentence, “a number minus 6 is less than 2” into an inequality. “A number” states the presence of an unknown variable. Let us denote that by . Aftersubtracting6from ,thisexpression should be less than (denoted by inequality sign <) 2. Now the expression becomes: –6 < 2 Similarly, mathematical expressions can be formed using differentinequalitynotationslikemorethan (>), less than (<), at most (≤), at least (≥), etc., as shown below.
+ 5 < 15 5 ≤25 − 5≥17 > 65
The sum of 5 more than the number is no more than 15
Five boxes of cereal costs at most 25 dollars
The difference between a number and 5 is at least 17
Jenny’s height is greater than 65 inches
E. Solving Equations Inordertosolvetheequations,itisimportanttomoveallvariablestoonesideusinginverseoperations (addition subtraction & multiplication division). It should be noted that applying inverse operations on both sides ofanequationkeepsthebalance.Forinstance,theequation canbe –3 = 7
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Chapter 2: Financial Mathematics
Overview Money is an essential componentofeverydaylife.Sometimesweworktoearnmoneyandsometimes money works for us, helping us earn more money. Financial mathematics deals with applies mathematicstoreallifesituations.Weallmakepurchasesofphysicalaswellasfinancialproductsand servicesandwiththehelpoffinancialmathematics,wecanbecomesmartconsumers.Themotivationof thiscourseisnotjusttoexcelinyoureducationalcareer,buttoequipyouwiththeskillstounderstand the daily transactions happening around you. You will become an intelligent consumer and saver by learning even the basic concepts underlying financial mathematics. This chapter will cover important concepts like percentage, discounts, markups, taxes, changes in percentage, profits and losses. Further, the financialtransactionstakingplaceinthefinancialmarkets are also taught in this lesson, which consists of simple interest, compound interest, effective rate of interest, continuous compounding, effective annual yield, and present and future values. Objectives
By the end of the chapter, you will be able to understand and apply the concepts of:
● Percentages and their usages ● Changes in percentages ● Different interest rates ● Compounding techniques ● Present and future value
A. Percent We are familiar with the word ‘percentage’andoftenuseittodescribeinterestrates,growthinsales, examscores,etc.Forexample,ifastudentscored85%ontheCLEPcollegemathematicsexamination,it would mean they got 85 questions correct out of 100: 1 8 0 5 0 % Butwhatdoespercentorpercentagemean?Percentisgenerallyusedtodescribeanyvalue‘outof100’, which means ‘divide by100’.Thedenominator,however,doesnotalwaysdefaultto100.Inthatcase, make the denominator100bymultiplyingthefactorbyascaler.Forexample,supposeRandallscored 15 out of 20, then the fraction becomes: 1 2 5 0
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Now, multiply both numerator and denominator by 5 so that the denominator becomes 100. 1 2 5 0 * 5 5 = 1 7 0 5 0 Letusconsideranothercase,whereyouhavescored65outof75.Nowinthiscase,multiplyanddivide the fraction by 100 and substitute the 100 in denominator as % sign and simplify the same: 6 7 5 5 * 11 00 00 = 6 7 5 5 * 100% = 86. 67% Sometimesweneedtofindthepercentageofaparticularvalue--forexample,calculatinga10%tipona restaurant check of $460. To do this, find 10 percent of $460. Mathematically, 1 1 0 0 0 * 460 = $46 B. Discounts, Markups, and Taxation We all are familiar with the terms like ‘discounts, markups, and taxation.’ Calculating these amounts makesuseofpercentageconceptsintheprevioussection.Forexample,yougotoagrocerystorewhere you see anofferthatifyouspendover$500,yougeta5%discount.If,forexample,youspend$570, then you can find the discount amount by finding 5% OF the total, that is: 1 5 0 0 * 570 = $28. 5 So, you pay 570 less 28.5 = $541.5 Alternatively,weknowthat100%ofavalueisequalto570.Sinceyougetadiscountof5%,youendup paying (100% - 5%) of the purchase price. That is: 1 9 0 5 0 * 570 = $571. 5 Just remember that a decrease in the price of an item (as compared to its list price) is termed as discount. Markupiswhenthepriceofanitemisincreasedfromitsbaseprice.Forinstance,aretailerpurchases itemsfromawholesalerormanufactureratwholesalepriceandthensellsthesameproductsatahigher price. The difference between the two prices is the ‘markup.’ For instance, an apparel outletsources mens denims from the wholesaler at $50, and then addstheirmarkupof60%beforesellingittothe customers. What is the retail price? Now, we can calculate the same in two ways. First, calculate the amount of markup and add to the purchase price. That is: = 1 6 0 0 0 * 50 = $30 Retail price = purchase price + mark up = 50 + 30 = $80 = 75%
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Chapter 3: Geometry
Overview Geometryisanessentialpartofmathematics,asitprovidesgeometricreasoningtoproblemsolving.We alldealwithdifferentgeometricshapesinourdailylives,likecircles,triangles,rectangles,squares,etc. Forinstance,yougofordailywalkinarectangularparkandyouneedtoseehowmanymilesyouwalk daily, so you can find out by learning about the perimeterofthatrectangle.Similarly,youwanttoget your windowpaintedwhichlookslikeasemi-circleonthetopofasquare,andthepainterasksyouto payhim$2.50persquaremeterofthatwindow.Sohereyouneedtobeawareabouttheareaoftriangle and semi-circle. Similarly, there are many more instanceswhereyoumakeuseofgeometricalshapes around you. The courseingeometrywillprovideyouwithbasicskillsinlines,triangles,quadrilaterals, and circles that will hone your problem-solving in dealing with different shapes. In this chapter, you will learn abouttheconcepts,classification,propertiesandtheoremsoftriangles, differentquadrilateralsandhowtheydifferfromeachother,parallelandperpendicularlinesinaplane
and circles. Objectives
By the end of the chapter, you will be able to learn the concepts and their application in:
● Triangles and their types ● Area and perimeter of triangles ● Quadrilaterals- classification, area, and perimeter ● Parallel and perpendicular lines ● Circles and its properties, circumference, and area ● Theorems of circles to measure different angles A. Triangles
Triangles can be defined as any geometric figure which has three angles and three sides. Before we move to understand the properties and application of triangles, let us understand different types of angles. Angles canbeclassifiedasacuteangles(thatmeasureslessthan90 0 ),rightangles(measures exactly90 0 ),obtuseangles(measuresmorethan90 0 ),andstraightangle(measuresexactly180 0 ).The shapes of these angles look like the ones given here:
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Based on these properties of angles, triangles are also classified as acute triangle (wherein all three angles of the triangle are acute), right triangle (one angle of the triangle is arightangle),andobtuse triangle (one angle of thetriangleisanobtuseangle).Itshouldalsobenotedthatthesumtotalofall threeangleofthetrianglesshouldalwaysbeequalto180 0 .Thedifferenttypesoftrianglescategorized on angles are given as follows:
Triangles can also be categorized on the basis of their sides, and are termedasscalene(noneofthe sides of the triangle are congruent or same), isosceles (two sides of the triangle are congruent),and equilateral (when allthreesidesofthetrianglearecongruent).Theclassificationoftrianglesbasedon sides is shown here:
Triangleinequalitytheorem statesthatthesumoftwosidesofanytrianglewillalwaysbegreaterthan the third side. For instance, a triangle cannot be formed if its sides are measured as 5cm, 7cm and 13cm, since adding two sides: 5+7 is not greater than the third side, that is, 13cm. Area of a triangle: The area of the triangle can befound by using the following formula: = 1 2 * * ℎ ℎ = 2 ℎ The perimeterofthetriangle canbefoundbyaddingthemeasureofallthreesidesofatriangle,thatis P = a + b + c Pythagorean Theorem: Thistheoremisapplicableonlytorightangletriangle.AsperthePythagorean Theorem,thesquareofthehypotenuseofthetriangleisequaltothesquareoftheothertwosidesofthe triangle. The sideoppositetotherightangleoftherightangletriangleisknownasthehypotenuse.In the given triangle,
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Chapter 4: Counting and Probability
Overview
We make use of counting and probability in our everyday life very often, like when we say thereisa likelihood of rain tomorrow. Similarly,weoftenmakeuseofwordslikelikely,probably,chance,etc.In such cases, we are not certain about the future happening but we are certain that it will happen. Countingandprobabilityhelpequipyouwiththebasicunderstandingofdeterminingchanceswhenwe deal withdifferentexperiments,likethelottery,winningatcasinos,playingcards,andotherforecasts. Finally, it facilitates healthy decision making in the light of uncertain future in different disciplines. In this chapter, you will learn the concepts and application of different counting techniques and probabilitysituations.Thechapterwillmajorlycoverthefundamentalcountingprinciple,permutations, combinations, and theories of probability. The theories and techniques presented in this chapter will also be applied in real life situations by way of different arbitrary examples. Objectives
By the end of this chapter, you will learn, solve and gain expertise in:
● Counting technique and rule ● Permutation and combination ● Probability with mutually exclusive and independent events ● Union and intersection of probabilities ● Expected value of payoffs ● Conditional probability A. Counting
Mathematicsinvolvestheuseofcountingandwemakeuseofdifferentcountingthroughoutourday.For instance,yougotoCocoberryFrozenYogurt,whereyoucanselectaflavorandtoppingsforyourself.The vendor asks you to choose between green apple, mango, strawberry, blackberry, blueberry, or black currantasyourbaseflavor.Notjustthis,nowyouhavetochooseonetoppingamongthe12optionsof fruits, nuts, and syrups; in this example counting can help to determine the total number of ways in which you can select your flavor. Similarly,wemakeuseofcountingineverysphereoflife.Itiseasytodetermineinsimplecaseswhere the options available are less, for instance, travelling between two cities involves three ways. The problem arises when the number of choices increases, like the case of Cocoberry Frozen Yogurt. Let use learn some simple rules of counting that helps to solve suchproblems.Beforemovingtothe rules, let us solve one simple problem of counting.
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Suppose Emily, who resides incityA,wishestogotocityCviacityB.Itisknownthattherearethree differentroutesbetweencityAandB,andthentherearetwodifferentroutesbetweencityBandC.Now we have to calculate how many options does Emily have? Let us understand this by drawing a tree diagram. Tree diagram refers to the graphical representation of all possible outcomes.
Intheabovecase,EmilyhasatotalofsixoptionstotravelfromcityAandCityCviacityB,thataread, ae,bf,bg,ch,andci.Hence,thereareatotalofsixoptions.Itwillbetoughtocountthetotalpossibilities if the sample space is much higher. Sample space refers to the set of all possible outcomes. B. Fundamental Counting Principle Ifoneeventoccursinmways,andanothereventoccursinnwaysaftertheoccurrenceofm,thenthe total number of ways in which the two events can occur is calculated as m X n ways. In the above example,EmilycantravelbetweencityAandcityBinthreepossibleways,andthenumberofwaysin which between city B and C istwo.Applyingthefundamentalcountingprinciple,thetotalnumbersof ways in which Emily can travel are 3 * 2 = 6 ways. Let us take a few more examples to gain expertise in the concept. Remember the Cocoberry Frozen Yogurtofferingthechoiceof6baseflavorsofyogurtand12optionsoftoppings?Wecancalculatethe totalnumberofpossibilitiesofdifferentyogurtsfromthestoreusingthecountingprinciple.Inthiscase, m = 6 and n = 12. Thus, we can say that there are 6*12 = 72 ways of selectingthefrozenyogurtat Cocoberry. Now,letusconsideranexamplewithabiggersamplespace.Inacountry,theautomobilesbearlicense plates with a combination of different letters and digits, such that the initial 2 positions are filled by letters and then the last three positions are taken by single digits. Now, what is the total number of possibilities of different license plates be made out of this? Weknowthat,thereare26letters(AtoZ)andatotalof10digits(from0to9).Thenumberofwaysin which first position of the licenseplatecanbefilledis26,thesecondpositioncanalsobefilledin26 ways,thethirdin10,thefourthin10,andthefifthpositionin10ways.Hence,thetotalnumberofways the license plate can be made is:
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