PEAULI PEAUIL PEALIU PEALUI PEAILU PEAIUL PEULIA PEULAI PEUALI PEUAIL PEUILA PEUIAL PELUIA PELUAI PELAUI PELAIU PELIAU PELIUA PEILUA PEILAU PEIALU PEIAUL PEIUAL PEIULA
APULIE APULEI APUELI APUEIL APUILE APUIEL APLIUE APLIEU APLUIE APLUEI APLEUI APLEIU APILUE APILEU APIULE APIUEL APIELU APIEUL APELIU APELUI APEULI APEUIL APEIUL APEILU
AUPLIE AUPLEI AUPILE AUPIEL AUPELI AUPEIL AULPIE AULPEI AULEPI AULEIP AULIEP AULIPE AUIPLE AUIPEL AUIEPL AUIELP AUILEP AUILPE AUEPLI AUEPIL AUELIP AUELPI AUEIPL AUEILP
ALPIEU ALPIUE ALPUIE ALPUEI ALPEUI ALPEIU ALUPIE ALUPEI ALUEPI ALUEIP ALUIPE ALUIEP ALIPUE ALIPEU ALIEPU ALIEUP ALIUPE ALIUEP ALEPIU ALEPUI ALEUPI ALEUIP ALEIPU ALEIUP
AIPULE AIPUEL AIPELU AIPEUL AIPLEU AIPLUE AIULEP AIULPE AIUPLE AIUPEL AIUEPL AIUELP AILUPE AILUEP AILPUE AILPEU AILEUP AILEPU
AIELUP AIELPU AIEPLU AIEPUL AIEUPL AIEULP
AEPLUI AEPLIU AEPILU AEPIUL AEPULI AEPUIL AEUPLI AEUPIL AEULIP AEULPI AEUILP AEUIPL AELPUI AELPIU AELIPU AELIUP AELUPI AELUIP AEIPLU AEIPUL AEIUPL AEIULP AEILPU AEILUP
UPALIE UPALEI UPAILE UPAIEL UPAEIL UAPELI UPLAEI UPLAIE UPLIAE UPLIEA UPLEIA UPLEAI UPILAE UPILEA UPIELA UPIEAL UPIAEL UPIALE UPELAI UPELIA UPEAIL UPEALI UPEIAL UPEILA
UAPLIE UAPLEI UAPELI UAPEIL UAPIEL UAPILE UALPIE UALPEI UALEPI UALEIP UALIPE UALIEP UAIPLE UAIPEL UAIELP UAIEPL UAILEP UAILPE UAEPLI UAEPIL UAELPI UAELIP UAEIPL UAEILP
ULPAIE ULPAEI ULPEAI ULPEIA ULPIEA ULPIAE ULAPIE ULAPEI ULAEPI ULAEIP ULAIPE ULAIEP ULIPAE ULIPEA ULIAPE ULIAEP ULIEPA ULIEAP ULEPAI ULEPIA ULEAPI ULEAIP ULEIPA ULEIAP
UIAPLE UIAPEL UAILEP UIALPE UIAELP UIAEPL UIPLAE UIPLEA UIPELA UIPEAL UIPAEL UIPALE UILPEA UILPAE UILAPE UILAEP UILEPA UILEAP UIEPLA UIEPAL UIELPA UIELAP UIEAPL UIEALP
UEPALI UEPAIL UEPLAI UEPLIA UEPILA UEPIAL UEAPLI UEAPIL UEALPI UEALIP UEAIPL UEAILP UELPAI UELPIA UELIPA UELIAP UELAPI UELAIP UEIPLA UEIPAL UEIAPL UEIALP UEILPA UEILAP
LPAUIE LPAUEI LPAEUI LPAEIU LPAIEU LPAIUE LPUIEA LPUIAE LPUAEI LPUAIE LPUEIA LPUEAI LPIAEU LPIAUE LPIEAU LPIEUA LPIUEA LPIUAE LPEIAU LPEIUA LPEUAI LPEUIA LPEAUI LPEAIU
LAPUEI LAPUIE LAPIEU LAPIUE LAPEIU LAPEUI LAUPIE LAUPEI LAUIEP LAUIPE LAUEPI LAUEIP LAIPUE LAIPEU LAIEPU LAIEUP LAIUPE LAIUEP LAEPUI LAEPIU LAEIUP LAEILU LAEUIP LAEUPI
LUPAIE LUPAEI LUPEAI LUPEIA LUPIEA LUPIAE LUAPIE LUAPEI LUAEPI LUAEIP LUAIPE LUAIEP LUIPAE LUIPEA LUIAPE LUIAEP LUIEPA LUIEAP LUEPAI LUEPIA LUEAPI LUEAIP LUEIPA LUEIAP
LIUPAE LIUPEA LIUEPA LIUEAP LIUAPE LIUAEP LIPAEU LIPAUE LIPEUA LIPEAU LIPUEA LIPUAE LIAPEU LIAPUE LIAUPE LIAUEP LIAEPU LIAEUP LIEPAU LIAPUA LIEAPU LIEAUP LIEUAP LIEUPA
LEPAUI LEPAIU LEPIAU LEPIUA LEPUIA LEPUAI LEAPUI LEAPIU LEAIPU LEAIUP LEAUPI LEAUIP LEUPAI LEUPIA LEUIPA LEUIAP LEUAPI LEUAIP LEIPAU LEIPUA LEIAPU LEIAUP LEIUPA LEIUAP
IPAULE IPAUEL IPAELU IPAEUL IPALEU IPALUE IPUALE IPUAEL IPULAE IPULEA IPUELA IPUEAL IPLAUE IPLAEU IPLUEA IPLUAE IPLEUA IPLEAU IPEALU IPEAUL IPEULA IPEUAL IPELAU IPELUA
IAPULE IAPUEL IAPLUE IAPLEU IAPELU IAPEUL IAUPLE IAUPEL IAULPE IAULEP IAUELP IAUEPL IALUPE IALUEP IALEPU IALEUP IALPEU IALPUE IAELPU IAELUP IAEUPL IAEULP IAEPLU IAEPUL
IUPALE IUPAEL IUPLAE IUPLEA IUPELA IUPEAL IUAPLE IUAPEL IUAELP IUAEPL IUALEP IUALPE IULPAE IULPEA IULEPA IULEAP IULAPE IULAEP IUELPA IUELAP IUEPLA IUEPAL IUEALP IUEAPL
ILPAUE ILPAEU ILPEAU ILPEUA ILPUAE ILPUEA ILAPEU ILAPUE ILAUPE ILAUEP ILAEPU ILAEUP ILUPEA ILUPAE ILUEPA ILUEAP ILUAPE ILUAEP ILEPUA ILEPAU ILEAPU ILEAUP ILEUPA ILEUAP
IEPAUL IEPULA IEPLAU IEPLUA IEPULA IEPUAL IEAPUL IEAPLU IEALPU IEALUP IEAULP IEAUPL IEUPAL IEUPLA IEUAPL IEUALP IEULPA IEULAP IELPUA IELPAU IELAPU IELAUP IELUAP IELUPA
EPAUIL EPAULI EPALIU EPALUI EPAILU EPAIUL EPULIA EPULAI EPUALI EPUAIL EPUIAL EPUILA EPLUAI EPLUIA EPLAUI EPLAIU EPLIUA EPLIAU EPILUA EPILAU EPIUAL EPIULA EPIALU EPIAUL
EAPULI EAPUIL EAPLIU EAPLUI EAPILU EAPIUL EAULIP EAULPI EAUPLI EAUPIL EAUIPL EAUILP EALPUI EALPIU EALUPI EALUIP EALIPU EALIUP EAIPLU EAIPUL EAIUPL EAIULP EAILPU EAILUP
EUPALI EUPAIL EUPLAI EUPLIA EUPILA EUPIAL EUALIP EUALPI EUAPLI EUAPIL EUAIPL EUAILP EULPIA EULPAI EULAPI EULAIP EULIPA EULIAP EUIPLA EUIPAL EUIALP EUIAPL EUILPA EUILAP
ELPAUI ELPAIU ELPUIA ELPUAI ELPAIU ELPAUI ELAPUI ELAPIU ELAIPU ELAIUP ELAUPI ELAUIP ELUPIA ELUPAI ELUIPA ELUIAP ELUAPI ELUAIP ELIPAU ELIPUA ELIAPU ELIAUP ALIUAP ELIUPA
EIPAUL EIPALU EIPLAU EIPLUA EIPULA EIPUAL EIAPUL EIPLU EIALPU EIALUP EIAULP EIAUPL EIUPAL EIUPLA EIULPA EIULAP EIUALP EIUAPL EILPAU EILPUA EILUPA EILUAP EILAPU EILAUP = 720
So the total number of arrangement for a six-lettered word whose letters are all different is 720.
Here is a table showing the total arrangements of words (where no letters are same) whose letters range from one to six.
It is apparent that as the number of letters in a word increase the number of different arrangements increase. When there is a two-lettered word the total arrangement is 2, which is worked out by doing 2 x 1 and 1 is also the total number of arrangement for a one-lettered. So I think that to find the total arrangements of a word you have to multiply the total number of arrangements of the word before it by the number of letters in that word. This theory can be confirmed by multiplying 24 (the total number of arrangements of a four-lettered word) by 5 (the number of letters in that word) which equals 120. But to find the total arrangements of a four lettered-word you have to multiply 1, which is the total arrangement of a one-lettered word by 2 and then multiply that by 3, which gives you 6. Then you multiply 6 by 4 to find the total arrangements of a four-lettered word. You can put all this together like this: 1x2x3x4x5 = 120.
So to find the total arrangements of any letter you have to start by multiplying from 1 to the number of letters of that word. So to find the total arrangements of a seven-lettered word you do this: 1x2x3x4x5x6x7, which equals 5040. This is called factorial.
Factorial is known by ! which can be found on a scientific calculator.
So the formula to find the total arrangement of a word, which has no repeated letters in it is n!.
Part 2
I will now investigate the total arrangements of words, which have some repeated letters in them. Then I will try and find a formula for finding the total arrangement.
First I will investigate the total arrangements of words, which have a letter repeated in it twice.
Two-lettered word with a letter repeated twice
AA = 1
There is only one arrangement for a two-lettered word, which has a letter repeated in it twice.
Three-lettered word with a letter repeated twice
RAA ARA AAR = 3
There are only three different arrangements for a three-lettered word, which has a letter repeated in it twice.
Four-lettered word with a letter repeated twice
EMMA EAMM EMAM
MEAM MAEM MAME
MMEA MMAE MEMA
AEMM AMEM AMME = 12
There are 12 different arrangements for a four-lettered word, which a letter repeated in it twice.
Five-lettered word with a letter repeated twice
SLEEP SLEPE SLPEE SPLEE SPELE SPEEL
SEEPL SEPLE SELPE SEPEL SEELP SELEP
LSEEP LSEPE LSPEE LPEES LPESE LPSEE
LEEPS LEESP LESPE LESEP LEPES LEPSE
PSLEE PSELE PSEEL PLSEE PLESE PLEES
PEELS PEESL PESEL PESLE PELSE PELES
ESPEL ESPLE ESEPL ESELP ESLEP ESLPE
EPSEL EPSLE EPLSE EPLES EPELS EPESL
ELSPE ELSEP ELPSE ELPES ELEPS ELESP
EELSP EELPS EEPLS EEPSL EESPL EESLP = 60
There are 60 different arrangements for a five-lettered word, which has a letter repeated
Here is a table showing the total arrangements of words, which have a letter repeated in it twice.
If you compare this with the total arrangements of words, which have no letters repeated in them you will notice that the total arrangement of a word, which has a letter repeated twice is half of the arrangement of a word, which has the same number of letters but has no letter repeated twice.
So to find the formula I would have to use the formula for a word, which has no letter repeated which is n! and then divide it by 2. So when you divide 24 (which is the total arrangement of LUCY) you get 12 (which is the total arrangement of EMMA).
So the formula to find the total arrangement of a word, which has a letter repeated in it twice is (n!)/2.
Now I will investigate the total arrangements of words, which have a letter repeated three times.
Three-lettered word which has a letter repeated three times
TTT = 1
There is only one arrangement for a three-lettered word, which has a letter repeated three times.
Four-lettered word which has a letter repeated three times
STTT TSTT TTST TTTS = 4
There are four different arrangements for a four-lettered word, which has a letter repeated in it three times.
Five-lettered word which has a letter repeated three times
ASTTT ATSTT ATTST ATTTS
SATTT STATT STTAT STTTA
TASTT TATST TATTS TSATT
TSTAT TSTTA TTSAT TTSTA
TTAST TTATS TTTAS TTTSA = 20
There are 2o different arrangements for a five-lettered word, which has a letter repeated three times.
Here is a table showing the total arrangements of words, which have a letter repeated three times.
If you compare this with the total arrangements of words which, have no letter repeated you will notice that the total arrangement of a word, which has a letter repeated three times is a sixth of the arrangement of a word, which has the same number of letters but no letters are repeated.
So the formula to find the total arrangement of a word which has a letter repeated three times is n!/6.
I will now investigate on the total arrangements of words, which have a letter repeated four times.
Four-lettered word which has a letter repeated four times
AAAA = 1
There is only one arrangement for a four-lettered word, which has a letter repeated four times.
Five-lettered word which has a letter repeated four times
RAAAA ARAAA AARAA AAARA AAAAR = 5
There are five different arrangements for a five-lettered word, which has a letter repeated four times.
Six-lettered word which has a letter repeated four times
RHAAAA RAHAAA RAAHAA RAAAHA RAAAAH
HRAAAA HARAAA HAARAA HAAARA HAAAAR
ARHAAA ARAHAA ARAAHA ARAAAH AHRAAA
AHARAA AHAARA AHAAAR AARHAA AARAHA
AARAAH AAHRAA AAHARA AAHAAR AAARHA
AAARAH AAAHRA AAAHAR AAAARH AAAAHR = 30
There are 30 different arrangements for a six-lettered word, which has a letter repeated four times.
Here is a table showing the total arrangements of words, which have a letter repeated in them four times.
If you compare this with the total arrangements of words, which have no letters repeated you will see that the total arrangement of a word, which has a letter repeated four times is a 24th of a word, which has the same number of letters but has no repeated letters.
So the formula to find the total arrangement of a word which has a letter repeated four times is n!/24.
I have found the formulas for finding the total arrangement of words, which have a certain number of repeated letters but I haven’t found a formula that can find the total arrangement of words, which have different numbers of repeated letters. So I ill try and find a formula that can work out the total arrangements of words whose number of repeated letters vary.
From the formulas that I have found, I have noticed that the number n! is divided by the result of doing factorial with the number of repeated letters. For example, if you want to find the total arrangement of a five-lettered word, which has two repeated letters, you will first do 1x2x3x4x5 which equals 120 and then do 1x2 (2 is the number of repeated letters) which is 2. Then you will divide 120 by 2 which equals 60 which is what I also got in the table.
This can be made more simpler like this: (5!)/(2!)
= (1x2x3x4x5)/(1x2)
= (120)/(2)
= 60
So the formula to find the total arrangement of a word whose number of repeated vary is (n!)/(r!) (r stands for the number of repeated letters in the word)
I will now investigate on the total arrangements of words, which have more than one type of letter repeated.
Four-lettered word which has two types of letters each repeated twice
RRAA RARA RAAR ARRA ARAR AARR = 6
There are 6 different arrangements of a four-lettered word, which has two different types of letters each repeated twice.
The formula for a four-lettered word, which has only two types of letters is n!/4.
You get 4 when you multiply 2! by 2! So I think that the formula to find the total arrangement of a word, which has two types of letters repeated two times or more is (n!)/(r!)(x!) (x stands for the other letter which is also repeated)
So to find the total arrangement of a four-lettered word, which has two types of letters repeated twice, you would do this: 4!
(2!)(2!)
= 24
4
= 6
But if there are more than two types of letters repeated all you do is add more factorials according the number of types of different letters that are repeated.
So to find the total arrangement of RRHHAA you will do this: 6!
(2!)(2!)(2!)
You have three factorials because there are three different types of letters repeated twice
= 720
8
= 90
So the total arrangement of RRHHAA is 90.
So the formula to find the total arrangement of a word, which has more than one type of letter repeated is (n!)/(Factorials) (the number of factorials depend on the number of different types of letters repeated)
Here are all the formulas that I have found from this investigation.
n! When there are no letters repeated.
(n!)/(r!) When there is a letter repeated 2 or more times.
(n)/(Factorials) When there are more than one type of letter repeated.
From this investigation I have found out many formulas, which will help me to find the total arrangements of different types of words. Finding the formulas weren’t too difficult but trying to put them into words was a bit hard.
