COMMON COUNTING CASES

FOUNDATIONSor numberofwaystochaos1elementfromtheaccumulationofmultipledisjointsets

1 TOTAL NUMBER OF ELEMENTSIN MULTIPLE DISJOINT SETS disjoint separate

noelementsincommon

Iset 11 t I set 21 t t set n Crueofsum or

Example

There are 80 chips salsa combinationsavailable at TraderJoe's There are21 chipssalsa combinations available at Fortherecord Imade

thesenumbersupCVS Nobrands are shared Therefore thereare 80121 101 chips salsa combinationsavailable across both TraderJoe's AndCVS ornumberofwaystochoose1elementfromset1 1elementfromset2 and1 element

fromsetn

2 TOTALNUMBEROF SET 1 ELEMENT imagineyou'refillingnslotswithanelementfromeachset

SET N ELEMENT TUPLESwheresetsaredisjointorwhererepetitionisallowed

1set 11 Iset21 K Iset n I cruleofproduct and

Example

If I have 5 shirts and 3 skirts there assumeeachoftheshirtsandskirtsisunique

are 5 3 15differentshirtskirt combosI can choose

3 TOTAL NUMBER OF ELEMENTSIN MULTIPLE MAYBE OVERLAPPING SETS

Iansetssetili Lift insetilt.E.IEYIinsetinsetkl infusioneoxyyigenofrin.cmD

ExampleIf there are 20 types of bagels at PoshBagel 20types of bagels at Houseof Bagels and 15 types of bagels which bagels

I i am

it

4 TOTAL NUMBER OF ENTITIESWITH ATLEAST ONE SOMETHING

subtractoutthenumberofLotottaehffies toitthatzfaootsoemmetheings elementsinthecomplementofthedesiredsetalternativelyyoucancountallthecases onesomethingExample twosomethings threesomethingsbutthisisoftenmoredifficult

The number of length 5 ABCDstrings doesnothavetobe atleastoneiwith at least one A is 45 35 youcanusethecomplementtickforanysetyou'retryingtocount

5 TOTAL NUMBER OF WAYS TOORDER N ELEMENTS

thinkof it asfilling n slotsn optionsforthefirstslot

n n l I n n 1 optionsforthesecondbecausenoreplacement andso onI E I

PERMUTATIONS AND COMBINATIONS

TOTAL NUMBER OF WAYS TOCHOOSE k FROM A SELECTIONOF N ELEMENTS WHERE

6 ORDER MATTERS AND orders

alwaysimagineyou'refillingTHERE 15 REPLACEMENT kslots

nowaskyourselfwhethernkis equivalentto

Example it ordermattersits orderdoesn'tmatterWe can make 53 three letter words oryou'recolorblind

fromthe letters ABCDE

7 ORDER MATTERS ANDTHERE IS NO REPLACEMENT

n Cn 1 Ln KHI

Cn Kj Pln K

ExampleThere are 7 6 5 4 ways thatwe can arrange 4 of 7 differentbooks into a pile

8 ORDER DOESN'TMATTER ANDTHERE IS REPLACEMENT

why basicallyyou'rethrowing

EYIYj K I t inEastinftp.a.saan.ceendi.niieednbthenumberofwayswecanorderKdartsand Ddividers isCktn1 dividedbyk becausetheorderofthedartsdoesn'tmatteranddin'dExample byCnD becausetheorderofthe

If a doughnut store sells 6dividersdoesn'tmatter

fora betterexplanationseethebelotypes of doughnuts and youwant sopostnotethattheyswitch k n

to buy 4 then assuming theyhave at least 4 of everydoughnutin stock there are 4 differentbags of doughnuts you can endupwith

9 ORDER DOESN'T MATTER ANDTHERE IS NOREPLACEMENT

k isthenumberofdifferentorderingsofn n l H n Kt1 kelements iforderdoesn'tmatterallofthepermutationsshouldbetreatedasthesamethereforewehavetoadditionallydivideatK byk toavoidcountingeachgroupofkelementsas k distinctpermutations

h npiccountseachcombinationk times

Elk L itinerant D.EEsasaiEeiengt