If I make the double- letter different then there will be the same amount of arrangements as Ben.
There is the same amount of arrangements as TJ when I make the ‘J’s different.
There may be a formula to work out the number of arrangements for a name with a certain amount of letters, without having to do all the working out. To see if I can find one I will put all my results into a table.
Table of Results.
I cannot see the formula yet, but I have noticed some patterns in the results. To find out the number of arrangements for a word with a double letter you have to half the number of arrangements a word has with single letters and the same number of letters (n=s/2(n being the double letter, s being the single letter)). To find how many arrangements there are you can also, multiply the previous number of arrangements by the number of letters in the word you want to investigate (n=xy (n being the no. of arrangements you want to find; x being the no. of previous arrangements and y being the no. of letters)). I cannot see the formula yet, but I have a good idea about how many arrangements you can have with five-letter names.
From the table I predict that a five-letter name with a double letter will have 60 arrangements, and a five-letter name will have 120 arrangements.
I will use the names Tasha and Rosie to investigate this.
I will now investigate the name Rosie.
I have found 120 different arrangements for the name Rosie.
I have also proved my prediction to be right.
You can also predict this in a different way. We know how many combinations are in a four-letter word, so you can also multiply the number of letters by the arrangements for a four-letter word. For example GEMMA, we know EMMA has 12 arrangements and if we multiply it by 5 we will get: 12*5=60, which is the correct number of arrangements.
I will now investigate the name Tasha.
I have found 60 arrangements for the name Tasha.
If I made the make the ‘A’s different there would be the same amount of arrangements as Rosie.
Table of Results.
I have now seen what the formula is. The formula that can be used is the factorial formula. To find out the number of single letters you use x! (X = the number of letters in the name), to find out the number of double letters you use (x!) / 2 (x = the number of letters in the name). This is because to get the number of arrangements for a three-letter name you have to multiply the one-letter arrangement by two, then you have to multiply it by three. These numbers are known as factorial numbers, the pattern goes like this:
1! = 1
2! = 1 x 2 =2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24 … etc.
We can also work out the nth term using this:
N! = 1*2*3*4*5……….(n–1)*n.
This works because (4-1)*4 = 12 and this is equal to 4! /2 is.
Using this formula I can find the number of arrangements for a six-letter name with a double letter and a six-letter name with single letters. To find out the double letter arrangement I would use (6!) / 2 which is 360, and the single letter arrangement is 6! which is 720.
I predict that if you have a triple letter in a name the number of arrangements will be a third of the double letter arrangement. Or to find the number of arrangements you could divide the single letter arrangements by 6.
This is what I think the results may be if a name had a triple letter:
To see if my predictions are correct, I will investigate names with a triple letter.
The first name I will use is a three-letter name –XXX
There is only one arrangement for XXX.
I will now investigate a four-letter word, with a triple letter; this name will be Lill.
I have found four arrangements for the name Lill.
I will now investigate a five-letter name with a triple letter; this name will be Bobby.
Table of Results, with predictions for a name with four-letters the same.
I think that a name with a name with four-letters the same will have a fourth of the arrangements that the triple letter name has. This would make it a 24th of the arrangement with no repeats (this is the same as 4!). This will make the formula x–24 (x = the number of arrangements for a name with no repeats). For a five-letter word with four letters the same the sum would be 120 divided by 24, which equals 5.
I will now work out arrangements for words with four-letters the same.
I will use BBBB
I have found 1 arrangements for BBBB.
I will now investigate ABBBB
I have proved my prediction to be right.
Table of Results.
I will now investigate the possible number of arrangements using X and A.
I will use X and A to investigate how many arrangements I can make from a double letter, double letter combination. I will use XXAA.
I have found 6 arrangements for XXAA.
I will now investigate a double letter, triple letter combination. I will use XXXAA
I have found 10 arrangements for XXXAA
I will now investigate a triple letter, triple letter. I will use XXXAAA
I have found 20 arrangements for XXXAAA.
Table of Results.
The numbers in blue are the number of arrangements that I have predicted using the formula. These should be accurate if my theory is correct.
I can see for a name with no repeated letters or names with repeated letters, is how you can work how many arrangements name has by:
N! N! N! N! N!
2 6 24 120
or you could write it like this,
N! N! N! N! N!
2! 3! 4! 5!
This works because 2! = 2, 3! = 6, 4! = 24 and so on. The formula for this is
N!
X!
Double letter, double letter = (4! / 2!) / 2!
Double letter, triple letter = (5! / 3!) / 2!
Triple letter, triple letter = (6! / 3!) / 3!
The formula for this could be
N! N = the number of letter.
X! Y! X = number of repeats.
Or Y = number of repeats (2).
(N! /X!) / Y!
To make sure this is correct I will see if it works by using the formula to check each set of results I have.
Double letter, double letter = (4! / 2!) / 2! = 6
Double letter, triple letter = (5! / 3!) / 2! = 10
Triple letter, triple letter = (6! / 3!) / 3! = 20
These are the same as my results so the formula is correct.
I will now prove that N! works
X!Y!
XXAA has 6 arrangements.
4! = 24 = 24 = 6
2!2! 2 x 2 4
XXXAA has 10 arrangements.
5! = 120 = 120 = 10
3!2! 6 x 2 12
XXXAAA has 20 arrrangements.
6! = 720 = 720 = 20
3!3! 6 x 6 36
In conclusion to my investigation, I have found out that a name with a double letter has half the arrangements of a name (with the same number of letters) with no repeated letters. A name with 3-letters the same has a 6th of the arrangements of a name with no repeats and a 3rd of a name with a double letter. I have also found out that a name with four letters the same has a 24th of the arrangements of a name with no repeats and a 4th of a name with a triple letter.
I have seen how the factorial numbers make it easier to work out the amount of arrangements a word has. I have also found which formula works to find out the number of arrangements names with double letter-double letter, double letter-triple letter and triple letter-triple letter. This formula is
N!
X!Y!
I have also found out that a five-letter word with all the letters different has the same amount of arrangements as a four-letter word multiplied by five. The formula would be 4! * 5, the overall formula is (N-1)*N
