CSC1300–ProblemSet5
1.ThemembersoftheVillanovaUniversityDanceTeamwillbeparticipatinginthe
NationalDanceAlliancecompetitioninDaytonaBeach,Florida.Expectationsare
runninghigh,sincethisawesometeamhasplacedtop10inDivisionIOpenDanceand
DivisionIHipHopoverthepastyearsinthenationalcompetition!Toprepareforthe
competition,oneoftheteammembersisponderingaparticularpiecethattheyare
performing,involvingsixmembersoftheteam:Amanda,Brianna,Christina,Daphne,
Ellie,Frances.
a)Howmanywayscanyoulineupthedancers?
b)HowmanywayscanyoulineupthedancersifDaphneorElliemustbeonthe
leftmostposition?
c)HowmanywayscanyoulineupthedancersifBriannaorAmandamustbeonthe
rightmostposition?
d)HowmanywayscanyoulineupthedancersifDaphneorElliemustbeonthe
leftmostpositionandBriannaorAmandamustbeontherightmostposition?
e)HowmanywayscanyoulineupthedancersifneitherDaphnenorElliecanbeinthe
leftmostpositionandneitherBriannanorAmandacanbeintherightmostposition?
2.Thedanceteamissettingupathree-membercommitteetoaddressdifficult
combinatorialquestions.Theyhavecalculatedthatthereare455waystoformthe
committee.However,theconcernisexpressedthatnotalltheteammembershave
takenCSC1300,sotheymightnotbepreparedtoserveonthiscommittee.Infact,ofthe
455possiblecommittees,220haveexactlyonememberwithadequatemathematical
background,66haveexactlytwo,andonly4haveallthreemembersuptothejob!
a)Howmanycommitteeshaveamajorityofmemberswithadequatepreparation?
b)Howmanycommitteeshavenobodywithadequatepreparation?
c)Howmanycommitteeshaveatmosttwomemberswithadequatepreparation?
d)Challenge:HowmanymembersdoestheVillanovaDanceTeamhave?Andhow
manyofthosedoyouthinkhavetakenCSC1300orsomeothercoursethatprepares
themtoaddressallthecombinatorialchallengesthatlieahead?
3.Considertheequation:
𝑥! + 𝑥! + 𝑥! + 𝑥! + 𝑥! + 𝑥! + 𝑥! = 15
a)Howmanyintegersolutionsarethere,if𝑥! ≥ 0,for1 ≤ 𝑖 ≤ 7?
b)Howmanyintegersolutionsarethere,if𝑥! ≥ 1,for1 ≤ 𝑖 ≤ 7?
c)Howmanyintegersolutionsarethere,if𝑥! ≥ 2,for1 ≤ 𝑖 ≤ 7?
VillanovaUniversityCSC1300www.csc.villanova.edu/~map/1300Dr.Papalaskari
4.Let’scountbananasplits.Theseareice-creamtreatsthathavethreescoopsofice
cream(twoorthreeofthescoopscouldbethesameflavor),threetoppings(twoor
threeofthetoppingscouldbethesameflavor),whippedcream(always),achoiceof
nutsornonuts,andachoiceofacherryornocherry,allplacedatoptwobananahalves.
Ifthereare18differentflavorsoficecreamand5choicesoftoppings,howmany
differentbananasplitordersarepossible?Notethatpeopledocarewhichtoppingsend
uponwhichscoops,sothepositionsofthescoopsshouldbelabeled.
5.Letussaythattwowordsareequivalentiftheyareanagramsofeachother.
a)Howmanyseven-letterwordsareequivalenttoTHOUGHT?(Notethatthewordsdo
nothavetoberealwordsormakesense.)
b)Isthisanequivalencerelationonthesetofallwords?
6.TheComputingSciencesdepartmentispreparingareportforaccreditationofthe
undergraduateCSCmajor.Dr.Casselneedsyourhelpindouble-checkingthereport.
Thereportstatesthat119studentstookCalculusIintheFall2015semester.The
reportnotesthat,inSpring2016,96ofthesestudentstookCalculusII,53ofthemtook
DiscreteMathematics,and39ofthemtookPhysicsII.Thereportalsosaysthat38ofthe
studentstookbothCalculusIIandDiscreteMathematics,31ofthestudentstookboth
DiscreteMathematicsandPhysicsII,32ofthestudentstookbothCalculusIIand
PhysicsII,and22ofthestudentstookallthreecourses.Youexaminethereportand
findadiscrepancy.Explain.
VillanovaUniversityCSC1300www.csc.villanova.edu/~map/1300Dr.Papalaskari