I now need to find a formula in order to make my future prediction far easier
After much research I have found out that the formula for a name with no repeats is N!
N! = The Number of letters X all the previous Number of letters I.E
To find the answer of 5 letter no repeat: N! = 5 x 4 x 3 x 2 x 1 = 120
To make sure ill try another one, 7 letters no repeats: N! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
My Formula works, I shall now investigate names with one letter repeated two, three or four times
JJ
A two letter name with one repeat = 1 arrangement
EDD
DDE
DED
A three letter name with one repeat = 3 arrangements
ANNE
NANE
NNAE
NEAN
NENA
NNEA
NAEN
AENN
ANEN
EANN
ENAN
ENNA
A four letter name with one repeat = 12 arrangements
Looking at my results below, I’m starting to see a pattern
1L 2L 3L 4L 5L 6L 7L
0 Repeats 1 2 6 24 120 720 5040
1 Repeat - 1 3 12 ? ? ?
It appears that the answers to names with one repeat is always half the amount of arrangements the names with no repeats have: I E (2/2=1) (6/2=3) and so on
From this information I can predict the following
1L 2L 3L 4L 5L 6L 7L
0 Repeats 1 2 6 24 120 720 5040
1 Repeat - 1 3 12 60 360 2520
The formula for names with one letter repeating once is ½ x N!
E.G For 6 letters with one letter repeating once: ½ x (6 x 5 x 4 x 3 x 2 x 1) = 360
For 4 letters with one letter repeating once: ½ x (4 x 3 x 2 x 1) = 12
The formula works!
I shall now look at names where two letters repeat once e.g Lulu
I will have to start at 4 letter names as you cant possibly have 2 letters repeating once in any smaller character name.
ANNA
NANA
NNAA
NAAN
ANAN
AANN
There are 6 possible arrangements for four lettered names with two letters repeating once
EDDIE
EDIDE
EDEID
EDDEI
EDIED
EDEDI
EEDDI
EEDID
EEIDD
EIDED
EIEDD
EIDDE
DDIEE
DDEIE
DDEEI
DIDEE
DIEDE
DIEED
DEEID
DEEDI
DEIDE
DEIED
DEDIE
DEDEI
IEEDD
IEDED
IEDDE
IDEDE
IDDEE
IDEED
There are 30 possible arrangements for a 5 letter name with two letters repeating once
Here are all my results I have found so far in a table:
In this table 1 repeat means there is one letter that was repeated. I.E Aandy However 2 repeats means there were two letters that each repeated once I.E. EEDDYA
Once again I have noticed that the arrangements for the names with two letters each repeating once are ¼ of the names with no repeats, and ½ of the names with one letter repeating once. Therefore the formula for a name with two letters repeating once is ¼ x N!
From this I can predict that the possible arrangements for a 5 letter name in which 2 letters each repeat once is 90. A 6 letter name in which 2 letters each repeat once has a possible 1760 arrangements.
I Can therefore work out a formula to suit all names in which a certain amount of letters repeat once.
The formula 1 divided by 2(to the number of repeats) x N!
Lets test the formula: For a 6 letter name in which 2 letters repeat once: 1/8 X ( 6 x 5 x 4 x 3 x 2 x 1) = 90
Once again I shall test it to confirm it: For a 5 Letter name with 1 letter repeating once:
½ x (5 x 4 x 3 x 2 x 1) = 60
The formula works
From this formula I now know that a 6 letter name with three letters all repeating once will have 45 possible arrangements.
Having found a formula for all names in which a certain amount of letters repeat once I shall now look at certain letters that repeat twice or three times and so on.
PQMMM
PMQMM
PMMQM
PMMMQ
QPMMM
QMPMM
QMMPM
QMMMP
MPQMM
MPMQM
MPMMQ
MQPMM
MQMPM
MQMMP
MMPQM
MMQMP
MMMPQ
MMMQP
MMPQM
MMMQP
5 letter word, One letter repeating three times, 20 arrangements
SMMM
MSMM
MMSM
MMMS
4 letter word, One letter repeated three times, 4 arrangements
I shall now compare these results to my other ones, however, I will only copmpare them to my results for 0 repeats and one letter repeating once as it is far clearer than if I also compare them to my results for two, three letters repeating once.
Table of Results
I have noticed that the answer to the 5 letter name in which the same letter is repeated twice, is a third of the answer to the 5 letter name in which the same letter is repeated twice. I therefore know that the other answers to names in which the same letter repeats three times will also be a third of the answers of the names where a certain letter repeats twice.
E.G.6 letter name in which the same letter repeats three times is a third of the 6 letter name in which the same letter repeats twice. I can therefore fill in the missing answers
From this I also know that the names in which the same letter repeats 4 times are going to be a quarter of the answer to the names in which the same letter repeats three times, I can now complete my table
Table of Results
I shall now try and find a formula which covers this:
n! / (n! of letters repeated)
I.E for a 7 letter name in which the same letter repeats 5 times; 7x6x5x4x3x2x1 / (5x4x3x2x1) = 42 It works!
for a 5 letter name in which the same letter repeats 4 times; 5x4x3x2x1 / (4x3x2x1) = 5 it works however I need to word the equation in a far more simplified way
Therefore the formula is N! % R!
In this equation N = No of letters in name
R = No of times a certain letter repeats in a name
I will finish this coursework by trying to find a suitable formula to suit every type of name;
To begin I will need to put every single result I have received in to one big table:
* The numbers with question marks next to them are predictions based on other results I have found. They are correct though as they fit with all the other pieces of information*
It is clear that the answers to names with two letters repeating N times will always be half of the answers where the name has 1 letter repeating N times
E.g. where a 6 letter name with 3 of one letter in it will have twice as many possible arrangements than a 6 letter name with 3 of two letters in it. This will help in finding a suitable formula as 2L is always going to be half the answer of 1L i.e. 2L = ½ N when 1L = N
Formula: l! % x! = possible arrangements (A)
(a) represents the total arrangements
(l) represent the number of letters in the name
(x) represent the numbers of the same letter in the name
(I) represents the recipicle
For example:
for 2 of the same letter in the name the formula is a=li/xixi
for 2 pairs of different letters in name the formula is a=li/x1ix2i
for 3 of the same letter in the name the formula is a=li/xixixi
for 3 pairs of different letters in the name the formula is a=li/x1ix2ix3i.
For example:
xxyy
the arrangement for this is a=(4*3*2*1)/(2*1*2*1)=6
xxyyy
the arrangement for this is a=(5*4*3*2*1)/(3*2*1*2*1)=10
xxxxxxyyyyyyyyyy
the arrangement for this is:
a=ni/x1ix2i=(16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(10*9*8*7*6*5*4*3*3*2*1*6*5*4*3*2*1)=8008
Use this formula, we can find out the total arrangements of all numbers and letters.
