I have found an equation, which will tell you the number of letter combinations in each word (except EMMA), it is based on the following idea:
In the word FRED for example, there are four letters. When rearranging the letters, there are four possibilities for where the first letter could be placed, and for each of those four possibilities, there are then another three possibilities for where the remaining three letters can go. This means in total so far there are 12 (4x3) possibilities so far.
For each of these 12 possibilities there are then another two possibilities for where the last two letters can go, that means we now have 24 possible combinations (4x3x2). Finally for each of these 24 possible combinations there is only one place for the last letter to go (4x3x2x1 in total). This idea is represented by a exclamation mark called a factorial.
The equation which represents this idea is: n! = a
Where n = the number of letters in the word
a = the number of different arrangements.
This equation works for the name LUCY and all the names in the table. However if we look back to the beginning of the investigation, we will see that it will not work for the word EMMA. In theory EMMA does have 24 combinations, because there are two instances of the letter M. This is easier to see if we look at the name EMMA like this:
EM1M2A
EM2M1A
Now we can see that the two EMMA’s are in fact different permutations. However I do not have an equation to find the amount of letter combinations if we ignore duplicate instances of the same letter and consider them as interchangeable.
After finding out the factorial equation I studied the different letter combinations in groups of letters that had repeated characters.
For example
XYZZZ
XZYZZ
XZZYZ
XZZZY
YXZZZ
YZXZZ
YZZXZ
YZZZX
ZXYZZ
ZXZYZ
ZXZZY
ZYXZZ
XYZZZ
ZYZZX
ZZXYZ
ZZYZX
ZZZXY
ZZZYX
ZZYXZ
This is a 5 letter word, with 3 repeated letters and 20 combinations
Results Table
From these results I can derive an equation. In the word LUCY there are 24 different combinations, however in the word EMMA there are 12. LUCY has 12 more combinations. Looking at the table we can see that when there is more than one instance of the same letter than the total number of combinations is reduced, therefore to find out the number of combinations in EMMA we need to reverse the process for finding more combinations.
N! / P! = A
Where n! = the number of letters in the word
p! = the number of letters the same
From the two formulas I have created I can now work out the number of letter arrangements in both LUCY and EMMA. LUCY I can find, by using the first, and EMMA by using the second. Also I can find out the number of arrangements of the other names I have selected.
However I have not fully completed my investigation, as I still cannot apply either of my formulas to a letter combination, which has more than one group of repeated letters. For example XXZZ, has four repeated letters in two different groups and one non-repeated one:
XXZZ ZZXX ZXZX
XZXZ ZXXZ XZZX
It has six different combinations.
Also look at the word: XXXYY
XXXYY XXYXY
XXYXX XYXYX
XYXXY XYYXX
YYXXX YXXXY
YXYXX YXXYX
It has eight different combinations
In my last equation I divided the number of letters by the number of permutations of the repeated letter, I believe if there is more than one group of repeated letter you merely have to multiply the number of permutation for each:
N! / (P! x Q!) = A
Where
N = Number of Letters
P and Q = different groups of repeated letters
A = Amount of combinations
If then you had a row of letters that were in unspecified amounts such as:
XX …… XXYY …… Y
Then you would simply apply the above formula, so :
N! / (X! x Q!) = A
Where
N = Number of Letters
X and Y = different groups of letters
A = Amount of combinations
