Arrangements For Emma:
There are 12 possibilities for the name Emma but if you note there are 4 total letters and 3 different.
Arrangements for Lucy, note which are all different letters:
There are 24 different possibilities in this arrangement of 4 letters all different. Double the amount as before with Emma’s name, which has 4 letters and 3 different. I have noticed that Lucy there are 6 possibilities beginning with each different letter. For example there are 6 arrangements with Lucy beginning with L, and 6 beginning with u and so on. 6*4 ( the amount of letters) gives 24.
What if there were 4 letters with 2 different?
Arrangements for aabb:
There are 6 arrangements for aabb. From 2 different letters to all different in a 4 letter word. I have found a pattern of 6, 12 and 24.
As I have found that there were 24 arrangements for a 4 letter word with all different letters and that there were 6 ways beginning with on of the letter I predict that there will be 120 arrangements
for lucyq, 24 for a, 24 for b, 24 for c; ect. 120 divided by 6 ( number of letters) equals 24. Previously in the 4 letter word, 24 divided by 4 equals 6, the number of possibilities there were for each letter.
For convenience sake I have used lucy and put a q in front of it to show that there are 24 different possibilities with each letter of a 5 lettered name being all different.
I can see that the numbers of possibilities for different arrangements are going to dramatically increase as more different as more letters are used. So as a general formula for names with x number of letters all different I have come up with a formula. With Lucy’s name; 1*2*3*4=24. With qlucy; 1*2*3*4*5=120. However this is expressed as a factorial. There is a button feature generally showing as an exclamation mark. All it does is save the time of having to put in to the calculator 1*2*3*4*5*6*7…. Etc. You just put in the number and press factorial and it will do 1*2*3… until it gets to the number you put in. If I key in (lets say the number of letters all different) factorial 6 I it gives me 720, which makes sense because 720 divided by 6 equals 120 which was the number of arrangements for a 5 letter word and it continues to fall in that pattern.
Total letters (all different) Number of arrangements
1 1
2 2
3 6
4 24
5 120
6 720
I have explained the pattern of general x lettered words. What shall I do if there are repeat letters? Like Emma; it has 4 letters but 2 of which are the same. 4 factorial equals 24, but I could only find 12, which means that there more to it that just factorial in that way.
To make an easier explanation I will use x’s and y’s (any letter) instead of letters.
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
There are 6 arrangements. What if I had xxxyy?
X=a, y=b
Aaabb aabab aabba ababa abaab
Abbaa bbaaa baaab babaa baaba
There are 10 different arrangements for this instance.
What If I had xxxxy?
X=a, y=b
Aaaab aaaba aabaa
Abaaa baaaa
There are 5 different arrangements for this instance.
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 withy, 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. I had a go at finding a logical universal general formula. I came up with; the number of total letters facxtorial, divided by the number of x’s, y’s ect factorised and multiplied.
Formula=
N!/x!y!
For example:
A five letter word like aaabb; this has 3 a’s and 3 b’s (3 x’s and 2 y’s)
So: 1*2*3*4 / 1*2 * 1*2=24 / 4=6 !!!
A five letter word like aaaab; this has 4 a’s and 1 b (4 x’s and1 y)
So: 1*2*3*4*5 / 1*2*3*4 * 1=120 / 24=5 !!!
Five letter words like abcde; this has 1 of each letter (no letters the same)
So: 1*2*3*4 / 1*1*1*1*1*1 =24 / 1=24
All these have been proved in previous arrangements. This shows that my formula works!!!!!
The National lottery can be shown as 49*48*47*46*45*44=
In the National Lottery you can get the numbers in any order so therefore you divide by 6! This can be expressed as:
N!/Nr(N-r)! or Ncr.