Name: Ian
Total number of letters: 3
Previous number of combinations: 2
3 X 2 = 6
Name: Lucy
Total number of letters: 4
Previous number of combinations: 6
4 X 6 = 24
This means that I can work out the total number of combination by factorial notation. Factorial notation is a number multiplied by the previous consecutive numbers:
E.g. 5! = 5 x 4 x 3 x 2 x 1 5! = 120.
Factorial notation is symbolised using an exclamation mark!
I realised this is because if I could find the total number of combinations by multiplying the total number of letters by the previous number of combinations it was the same as multiplying the total number letters by its previous consecutive numbers Factorial Notation.
The formula to find out the total number of combinations for any name when all the letters are the same is:
Number of combinations = Total number of letters Factorial
C = t!
I have find out I can do it in another method as well using numbers
LUCY=24 ARRANGEMENTS
L =1
U=2
C=3
Y=4
24! =1*2*3*4*
= 24
Table of Results
From the table of results I have found out that a 2-letter word has 2 arrangements, and a 3-letter word has 6.
Taking for example a 3-letter word, I have worked out that if we do 3 (the length of the word) x 2 = 6, the number of different arrangements.
In a 4 letter word, to work out the amount of different arrangements you can do 4 x 3 x 2 = 24, or you can do 4!, which is called 4 factorial which is the same as 4 x 3 x 2.
So, by using factorial (!) I can predict that there will be 40320 different arrangements for an 8-letter word.
Why this Formula Works
This formula works on the simple principle that once a letter is selected, the next selection will have 1 less choice, and so on.
For example: Lucy has 4 different letters.
Out of the 4 letters one letter has to be chosen to be the first letter, the second letter will be one of 3 possibilities because one has already been chosen (4 – 1 = 3). The third letter will be chosen out of 2 possibilities and the forth there will only be 1.
So then the possibilities have to multiplied together to achieve the total number of combinations.
4 x 3 x 2 x 1
So the original number of letters is the number of letters factorial.
Now I am going to investigate the number of different arrangements in a word with 2 letters repeated
EMMA
ARRANGEMENTS IN EMMA
EMMA EMAM EAMM MMAE
MMEA MEAM MEMA MAME
MAEM AMME AMEM AEMM
4-letter word, 2 letters repeated, 12 different arrangements.
Total number of arrangements is 12.
Having found a formula for Lucy when all the letters are different, I am now going to try to develop the formula for Emma, when 2 of the letters are the same.
If you look at four-letter word Lucy got 24 arrangements, but if you look at the name Emma it got four letters as well but it got 12 arrangements. This is because it got 2 letters the same.
I looked at other names with different total numbers and put the results in the table below.
Because the formula must be based on the same principle as the one for all letters different, I am going to start by working out the number of letters factorial, and look for a relationship between this number and the total number of combinations.
The difference between the total number of combinations and the total number of letters factorial is always total number of letters factorial divided by 2.
Therefore the formulae for the total number of combinations when 2 letters are the same is:
Combinations = total number of letters factorial / 2
C = t! / 2
For Emma
C = 4! / 2
C = 12
This means I can predict the total amount of combinations for the name Jenny.
Combinations = total number of letters factorial / 2
Combinations = 5! / 2Combinations = 60
Now I am going to look at different names with different number of repeated words
Arrangements for xxyy:
To make it a bit easier instead of using letters as such I will use x's and y's (any letter). I will start with xxyy:
Arrangements for xxyy:
x=a, y=b
This is a 4-letter word with 2 different. I have done this with;
aabb abab baab
aaba baba bbaa
As you can see there are 2 x’s and 2 y’s. There are 6 arrangements, 3 starting with each letter.
Arrangements for xxxyy:
x=a, y=b
aaabb aabab aabba ababa abaab
abbaa bbaaa baaab babaa baaba
There are 10 arrangements. There are 6 starting with x and 4 starting with y.
Arrangements for xxxxy:
x=a, y=b
aaaab aaaba aabaa
abaaa baaaa
There are only 5 arrangements. There are 4 starting with x and 1 starting with y.
Because there are two sets of same letters the previous formula cannot be used. However I have come up with another formula:
x!
(y! * z!)
x = total number of letters, y = number of repeated letters (1), and
z = number of repeated letters (2).
If I go back to xxxyy; there are 3 x's and 2 y's in a total of 5 unknowns. As each letter has its own number of arrangements i.e. there were 5 beginning with x, and 5 beginning with y, I think that factorial has to be used again. Also in a 5-letter word there are 120 arrangements and 24 arrangements (120 divided by 5) for each letter. As there I a divide issue involved I had a go at trying to work out a logical universal formula. I came up with; The number of total letters factorial, divided by the number of x's, y's ect factorised and multiplied.
For examples:
A five letter word like aaabb; this has 3 a's and 2 b's. So: 1x2x3x4x5 / 1x2x3 x 1x2 = 120 / 12 = 10
A four letter word like aabb; this has 2 a's and 2 b's
So: 1x2x3x4 / 1x2 x 1x2 = 24 / 4 = 6
A five letter word like aaaab; this has 4 a's and 1 b
So: 1x2x3x4x5 / 1x2x3x4 x 1 = 120 / 24 = 5
Five letter words like abcde; this has 1 of each letter (no letters the same)
So : 1x2x3x4 / 1x1x1x1x1x1 = 24 / 1 = 24
All these have been proved in previous arrangements. This shows that my formula works.
