E.g.
Judging from the names investigated so far it is apparent that if a name with five letters is investigated it will have 120 different arrangements.
E.g.
-
CAINE CAEIN CIEAN CNEIA CENIA
- CAIEN CIANE CIENA CNIEA CENAI
- CANEI CIAEN CNAIE CNIAE CEIAN
- CANIE CINEA CNAEI CEAIN CEINA
- CAENI CINAE CNEAI CEANI …………..
The above arrangements show the first 24 arrangements for the five-letter name CAINE. The first 24 arrangements all
Have the letter C in the front and in this same way if I was to carry on every other letter would come to the front and therefore there would be 24 x 5 arrangements, which is 120.
If two of the letters in a name are the same then there cannot be a name with just one letter but if it has two letters then there is simply one arrangement.
If a name has three letters two of which are the same then there will be two arrangements as shown below:
If a name has four letters with two letters the same then it will have 12 arrangements (i.e. Emma) and if it has 5 letters of which two are the same then it will have 60 different arrangements:
-
GEMMA GAMME GEMAM
- GAMEM GMMEA GMMAE
- GMEAM GMAEM GMEMA
- GMAME GAEMM GEAMM ………
These are the first 12 arrangements of the name Gemma that has two letters the same. The above arrangements have the same letter in the front and therefore if I were to complete it, each other letter would come in front and that would mean that 12 would start with G, 12 would start with A, 12 would start with E, and 24 would start with M.
The tables on the next page show the number of arrangements for different amounts of letters, including names with repeats.
The table above shows the results for names when there are no repeating letters. At the moment the way I have figured out the number of arrangements for six letters is by multiplying the number of arrangements for five letters by six. (120 x 6 = 720) this sort of rule works all the way from the top.
I.e. 2 x 3 = 6 and 6 x 4 = 24.
Another way of working across it is that:
Number of arrangements for 1 letter = 1 x 1
Number of arrangements for 2 letters = 1 x 2
Number of arrangements for 3 letters = 1 x 2 x 3
Number of arrangements for 4 letters = 1 x 2 x 3 x 4
Number of arrangements for 5 letters = 1 x 2 x 3 x 4 x 5
Number of arrangements for 6 letters = 1 x 2 x 3 x 4 x 5 x 6
Etc….
If two letters in the name are the same then the table is slightly different but even then the figures are obtained in the same way as they were for the other table.
When the two tables are compared it can be seen that the values in the table for when two letters are same are half that of the ones that are in the table for when all the letters are different.
The results for the second table cannot be obtained by the method:
1 x 1
1 x 2
1 x 2 x 3
1 x 2 x 3 x4
My next step was to refer different textbooks and when I looked in an A-level statistics booked called T1 under the heading factorials its states that:
The multiplication of 4 x 3 x 2 x 1 can be written as 4! And is called 4 FACTORIAL. This mathematical notation can be generalised as:
This notation fits the first part of the investigation perfectly and therefore a generalised rule for finding the number of different arrangements for a name with n number of letters would be n!
Proof
Lucy has 4 letters and 24 arrangements. Four letters would mean that 4! would equal 24 and it does;
4 x 3 x 2 x 1 = 24
Charlot has seven letters and so to figure out how many arrangements my name can have, 7! Should be worked out: the answer is 5040.
To check that this is right we can also figure out the answer using the initial method, which was multiplying the previous term by n. If n = 7 and the previous term is 720 (the number of arrangements for six letters) 720 x 7 = 5040.
n! is therefore the correct General formula
My next task was to figure out a general formula for when two letters are the same. Because the values for this table are half that of the first table I knew that the rule would have something to do with division or halving.
In the end I came up with the idea of number of letters over the number of letters the same factorial.
I.e. 4!/2!
This rule works because 4 x 3 x 2 x 1 = 4 x 3 = 12
2 x1
This is the number of arrangements for four letters when two are the same.
We could also work out the number of arrangements for when there are seven letters two of which are the same.
e.g. Bhavika has seven letters two of which are the same (a). This name would therefore have 2520 arrangements because 7!/ 2! = 2520
We could work out this number using the initial method and to do that we would multiply 360 by 7, which equals 2520.
Another sort of name (more a word) that I investigated was SENSES. This is a good word because it has one letter that is repeated three times and another letter that is repeated two times.
The formulae to figure the number of arrangements for this is 6!/ 3!2! (3! For the three S’s and 2!for the two E’s and 6! For the total number of letters), that gives the answer 60. We can check this by writing the arrangements out:
-
SENSES NSSSEE NSSEES
- NSSESE NSEESS NEESSS
- NESESS NESSSE NESSES
- NSESSE NSESES
It is visible that the letter N comes in front ten times therefore the other 5 letters will come forward 10 times each and there will then be 60 combinations.
For the X’s and y’s the same method then applies:
Total number of letters!/number of times letter repeated! X number of times letter repeated!
For example: xxxxxyyyyyyy = 12! / 5!7! =792 arrangements
There is also an x! Button on most scientific calculators and this also helps a great deal!!!!!
