LUYC UCLY CYUL YLCU
LYUC ULCY CLYU YULC
LYCU ULYC CLUY YUCL
LCUY UYCL CULY YCLU
LCYU UYLC CUYL YCUL
There are 24 ways in which I could re-arrange the name ‘Lucy.’ I am now going to find out how there are 24 different ways in which I can re-arrange ‘Lucy.’
Below is a table of how many times you can re-arrange groups of letters which have less letters than in ‘Lucy.’
I have found out that that there is a formula for it, below is a table showing my method of finding the formula.
I have found out that when multiplying the group of letters by the previous number of arrangements, I get the number of arrangements of the one I require.
1 2 3 4 5 6
1 2 6 24 120 720
=equals
=formula
I have found out that when multiplying the group of letters by the previous number of arrangements, I get the number of arrangements of the one I require.
For e.g. if I want to find the number of re-arrangements for 7 letters, I will use the formula below.
Number of letters*previous number of re-arrangements
7*720=5040
- I will now show you the different ways in which ‘Emma’ can be written as.
EMMA MMEA AEMM
EAMM MMAE AMEM
EMAM MEAM AMME
MAEM
MEMA
MAME
There are 12 ways in which I can re-arrange the name ‘Emma.’ This is because there are 2 M’s. They may be the same letters but they are counted separately.
Finding the formula for ‘EMMA’ will be somewhat harder since it has ‘2 M’s.’ Firstly I will re-arrange groups of letters with 2 of the same letters and see if there is a link between them and ‘EMMA.’
I have found out that the formula for the above consists of factorial, for e.g. when you want 2 find out how many arrangements there are for 7, you would normally do
7*6*5*4*3*2*1 =5040
To find this using the factorial you would do:
Y = N!
2
Where ‘Y’ equals total number of arrangements and ‘N’ equals total number of letters.
I will now use totally different letters to see if the formula works.
AAB
BAB
BAA
I have been able to re-arrange this group of three letters 3 times. This is because there are 2 repeats. So when using the formula, I also get the same answer.
N!
2
Below I have typed out all the re-arrangements of AAAB:
AAAB
AABA
ABBA
BAAA
When using the formula:
N!
2
I get the answer 12, so that makes the formula incorrect.
I have found out that the formula is not correct for all groups of letters.
The formula for AAAB is:
Y = N!
r
In the formula, ‘r’ is the number of repeats.
With the formula I can work out any word, which has more than one set of repeats. I will try and find how many arrangements can be made in the name:
AABBCCDDEEFF
Below is what I substituted for the formula:
12!
(2!*2!*2!*2!*2!*2!)
The answer I got was 748440, using the formula is much easier than writing out all the possible arrangements.
I will now use totally different letters to see if the formula works
AAB
ABA I have been able to re-arrange this 3-letter word 3 times.
BAA
When using the formula, I get:
3! /2! = 3
AABB
BAAB
BBAA
When using the formula, I get
4! /3! = 4
Below is a table of other groups of letters I have re-arranged:
For the above groups of letters I used the formula:
Y = N!
r
For e.g. when finding how many re-arrangements ‘AABBCCDDEEFF’ has, I did:
12! / (2!*2!*2!*2!*2!*2!) = 7484400
The answers to these groups of letters are lower because there are a great number of repeats.
I will now find out how many times I can re-arrange names of places or words which are very long indeed.
Firstly I will re-arrange the longest word in the dictionary:
ANTIDISESTABLISHMENTARIANISM
Using the formula:
Y = N!
r
As there are
5 I’s
4 S’s
4 A’s
2 E’s
3 T’s
2 M’s
3 N’s
This is what my calculator display looked like:
28!
(5!*4!*4!*2!*2!*2!*2!)
My answer equals to: 3
This is because there are so many letters that are repeats.
I also thought of re-arranging the name ‘MISSISSIPPI’ seeing as it has many repeated letters in it.
MISSISSIPPI
There are:
4 I’s
4 S’s
2 P’s
I have found out that I can re-arrange the word ‘MISSISSPPI’ 34650 times. This is mainly because there are so many repeat letters.
Conclusion
I have found out that my hypothesis was incorrect. I said that ‘the more letters there are, the more re-arrangements there would be whether there is more than one of the same letters.’
This statement is however incorrect as I soon discovered when finding out how many times I could re-arrange the name ‘EMMA.’
I realised that even though the number of letter were the same, there were repeats so the amount of times they could be re-arranged would not be the same.
To conclude, I would like to point out that the number of letters in a word is not the only thing that determines the number of arrangements possible but also repeats, if there are any.
