1228BW12t1v1_sol.pdf

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Department
Mathematics
Course
Mathematics 1225A/B
Professor
Jennifer Mori
Semester
Winter

Description
SaturdayFebruary42012Mathematics1228BCODE111Page1Test1PARTA17marks11markInagroupof85studentsitisfoundthat60studentshaveapen50studentshaveapenciland25studentshaveapenbutnotapencilHowmanystudentshaveapencilbutnotapenA10B15C20D25E35SolutionLetUbethesetofallstudentsinthegroupDeneAbethesubsetcontainingthosestudentswhohaveapenandBbethesubsetcontainingthosestudentswhohaveapencilThenwearetoldccthatnU85nA60nB50andnAB25andweareaskedtondnABSincecnAnABnABweseethatcnABnAnAB602535candsincesimilarlynBnABnABwegetcnABnBnAB50351521markLetGbethesetofallstudentstakingGeographyMbethesetofallstudentstakingMathandPbethesetofallstudentstakingPsychologyAllisonistakingbothGeographyandMathbutisnottakingPsychologyWhichoneofthefollowingsetsisAllisonincAPMBGMCGMPDGMPEGPMPcSolutionAllisonisinsetsGandMbutisnotinsetPsosheisinPSincesheisnotinsetPsheisnotinPAforanysetAieisnotintheintersectionofanysetwithsetPThereforesheisnotinanyofPMPGMGMPorGPMPPGMAndsincesheisinbothcGandMthensheisinGMandthereforeisnotinGMButthensincesheisinGMsheisalsoinGMAforanysetAieisintheunionofthissetsheisinwithanyothersetandthusisinGMP31markFindthenumberofdierentwaystogive45centsinchangeusingquarters25centsdimes10centsandornickels5centsHintConsidergivingcoinsoflargervaluebeforethoseofsmallervalueA2B3C6D8E9SolutionAtreeisusefulhereTherstlevelofthetreewouldbeforthedecisionhowmanyquartersaregivenwithchoices0or1ThesecondlevelofthetreeisforthedecisionhowmanydimesaregivenThechoicesheredependonwhetherornotaquarterwasgivenIfyesthenthereis20centslefttobemadeupso0or1or2dimesmightbegivenButifnoquarterwasgiventhereisstill45centstobegivenso0or1or2or3or4dimesmightbegivenFinallythethirdlevelofthetreeisforthedecisionandnowhowmanynickelsshouldbegivenForeachofthesecondlevelbranchesthereisonlyonethirdlevelbranchbecausealloftheremainderoftheoriginal45centsmustbegiveninnickelsForinstanceifonequarterand2dimeshavebeengiventhen0nickelswillbegivenorif0quartersand3dimeshavebeengiventhen3nickelsmustbegivenSincethereare8secondlevelbranches3growingfromthe1quarterrstlevelbranchand5growingfromthe0quartersrstlevelbranchtherewillbe8thirdlevelbranchesgiving8terminalpointsinthetreeThereforethereare8dierentwaystogive45centsinchangeusingquartersnickelsanddimesAndwedontneedtodrawthetreeButyouprobablydid
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