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Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   u,-.)*+%'153 +]+c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Question:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Howmanywaysaretheretostacktheblocksshowntotheright?Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Answer:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Thereare54321=120waystobuildsuchastackLc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..InGeneral:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Thenumberofwaystoarrangendistinctobjectsisn!=123nWPCxXX81%XX81%&3'XXXXLLxx9 " , ,W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   " ,.'13 `!: Z+`c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howmanywaysaretheretoarrangetheobjectsontheright?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Theanswerisnot4!,becausetheobjectsarenotalldistinctLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..WeneedawaytodealwithrepeatsLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Ourtrickisto:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..temporarilymakethemalldistinctM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..thencorrectforovercountingWPCt XX81%XX81%s3'XXXXLLxx9j" WWW @@&d9 Z6Times New Roman Regular +ނW ss&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   W-.)*+%'153 C:W%Mc&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0L..Herearethe24wayswithdistinctcolorsLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0L..TheyarenotalldistinctwiththeoriginalcolorsWPCXX81%XX81%s3'XXXXLLxx9j"nR/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ArrangingObjectswithRepeats.+%19׿3aQ `fR//RWPCXX81%XX81%3'XXXXLLxx9j"u./"u./"W xx&d9 Z6Times New Roman Regular +YW &d9 Z6Times New Roman Regular +YW &d9 Z6Times New Roman Regular +YW kk&d9 Z6Times New Roman Regular +YW kk&d9 Z6Times New Roman Regular +YW kk&d9 Z6Times New Roman Regular +Y)?  Z9׿aQ 4 @ @ @ @ @ @ f" dddddd$   u./"-.)*+%'53Y +.:/c&d9 Z6Times New Roman Regularckkk        @Y  Q Qc&d9 Z6Times New Roman RegularcxxxHowcouldwehavepredictedthateachgroupwouldhave4arrangementsinit?f  c&d9 Z6Times New Roman RegularcxxxWewouldthenknowthat24/4=6wastheanswerwithouthavingtodrawallthepictures!WPCXX81%XX81%3'XXXXLLxx9j"AR/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#%`L $ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc PredictingtheNumberineachGroup.+%19׿3aQ `R//RWPC*XX81%XX81%3'XXXXLLxx9j E0E0W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   E0-.*+%'153 GE:0Zec&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..HereisaredrawingofthepurplegroupLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howcouldwehavepredictedthattherewouldbe4arrangementsinsidethisgroup?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Therearetwo blueblocks.c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Theycanbearrangedin2!ways.Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Therearetwo redblocks.c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Theycanbearrangedin2!ways.Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Sothisgroupwillcontain2!'2!=4arrangements(bytheproductrule)Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Similarly,everygroupwillcontain4arrangementsWPCXX81%XX81%3'XXXXLLxx9j"2 R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y $ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc InsidethePurpleGroup.+%19׿3aQ `R//RWPCXX81%XX81%3'XXXXLLxx9("2(R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y 6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc HowManyineachGroup?.+%19׿3aQ `R// R WPCm XX81%XX81%3'XXXXLLxx9("zhm+zhm+W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   zhm+.*+'13 0h*c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..HereisanothertowerofcoloredblocksLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Supposeweassigndifferentcolorstoeachblock.Howmanyarrangements?WPCXX81%XX81%3'XXXXLLxx9("dtz +z +W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   z +-.)*+%'13 0 \* c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Samequestionforthetowerofblockstotheright:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Whenweassigndistinctcolorsandthenregroup,howmanywillbeineachgroup?M..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Didyousay3!'2!?WPCXX81%XX81%3'XXXXLLxx9("2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y `e6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc HowManyineachGroup?.+%19׿3aQ `fR// R WPC,XX81%XX81%3'XXXXLLxx9("Fgz/yz/yW @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   z/y.*+'13 e05/&c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howmany anagramsarethereofeachofthefollowingwords?c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..READc&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..4!=24c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..REEDc&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..4!/2!=12c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..DEEDc&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..4!/(2!'2!)=6c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..EPEEc&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @