A M
M M
A
M M
These different colours indicate that each colour paired equal to one. So each two boxes are in one colour and there are three different colour , so three different arrangements.
EMMA indicates one.
So the tree diagram above shows three different arrangements starting with one particular letter “E”. So four letters equal to 4*3=12 arrangements.
The total number of different arrangements for the word the word LUCY is 24 where as for EMMA is 12.however both of these words have the same amount of total letters. We can see this above that this happens because the name EMMA has the same letter “M” twice. Therefore as I mentioned before every time we counts the letter” M” i.e. EAMM, EAMM) as we are swapping the letter “M” we are loosing one arrangement (one chance) which originally should count as two.
Part 3
In this part I am going to look at the different arrangements of various groups of letters (words or names)
First of all, I am going to consider words or names with all letters different.
- A=1 Arrangement
- AB=2 arrangements
- ABC BAC CAB
ACB BCA CAB
=Six arrangements for three different letter word
C B
A
B C There are three letters all together and here I have used “A” as my first letter and as you can see there are two arrangement which starts with “A”. Therefore there are three different letters and two arrangements for each letter (starting letter)
3*2=6 arrangements
First of all we have got three possibilities “A, B, C” and then two possibilities “B or C”. We can see this in the tree diagram shown above.
Therefore 3*2*1=3!
- ABCD
There are four letters altogether and as I said earlier in part one that four different letters word have 24 arrangements by using LUCY, the same thing is happening in this four different letters word “ABCD”.
Having noticed that the number of different arrangement for a word with three different letters is 3! And that for a word with four different letters is 4!; now I can predict that the arrangements for a word with five letters will be 5*4*43*2*1=5!
A word with five different letters
There are 24 arrangements starting just with one letter “A”. therefore there are five letters altogether, so 5*24=120
Consequently there are 120 arrangements altogether.
51 = 5*4*3*2*1
= 120 arrangements.
Now I am going to draw a table of results and see whoat relationship between them.
Table 1
5*24 =120 (cross multiple)
I have noticed that for the word with all letters different, the factorial of the number of the letters in the word is equal to the number of different arrangements.
Let “N”=number of letters in the word
“A”= number of different arrangements
Therefore N! =A
For example: a three letter word with all letters different
Prediction: for the word with six letters different.
Using pattern noticed
1) N*(N-1) =A
6*5! =6*120
=720
Using the formula
- A=N!
A=6!
A=720
The calculations using formula and the pattern noticed both agree.
I am going to investigate words with two letters the same.
- HH – this has one arrangement
- TOO
OTO -this has three arrangements
OOT
- A word with four letters with two letters the same will give the same number of arrangements as EMMA i.e. 24 arrangements
- The arrangement for a word with five letters the two letters the same is shown on the next page. I have used the word “PAPER” and I have numbered each of the letter “p “to make them different from each other.
This is to enable me to work out the total number of arrangements when all the letters are different and to compare it when two of the letters are the same.
There are 60 different arrangements altogether, however
PAPER
PAPER 1 arrangement
We can see two arrangements, when we swap that “P”,”P” but as they look the same there are 120 arrangements when we use the P as P AND P. however as they look the same we divide 120 by two which is equal to sixty arrangements.
Now I am going to put the results on the table and see what relationships between them.
The word with two letters the same
Table 2
5*12=60 (cross multiple)
Number of different arrangements for the word with 2 letters the same equal to the factorial of the number of letters in the word divided by two
Let “N”=number of letters in the word
“A”= number of different arrangements
Therefore A=N! /2
For example: a three letter word with five letters and two letters the same.
A=5! /2
120/2 =60
I have also noticed that if we multiple for example
The number of letters “n” in the word * the number of different arrangements of the previous number
2
Gives the number of arrangements for “n2 of letters
i.e. N*(N-1)/2=A
5*(5-1)/2
5*4! /2
5*24/2
120/2=60
Prediction:
Testing the formula
Let’s take “TOO” and check my formula
So3! /2=6/2=3
N*(N-1)/2
3*(3-1)/2
3*2! /2
6*2=3
We can see the same results on the table before I work out the formula.
Therefore this proves that my formula is correct and it would work for any number of letter with two letters the same.
Now I am going to quickly look at the words with three, fur or five letters the same.
“MUMMY”
M Y
M Y M
M Y
U M Y M 3 different arrangements.
M M
M
M Y M M
M
Y
Same arrangements however there are six arrangements.
- MMYUM
- MMMUY
- MMYMU
- MMUYM
- MMMYU
- MMUMY
There should be 3 different arrangements.
Therefore 3+6+3=12 different arrangement starting just with “m” and now we got four more letters (starting letter e.g.-U, M, M, and Y). However we can’t do it again with (start) M and M because it is going to be same. So we now start with other two letters. (U, Y)
Y
M
M Y M
M Y
M M Y M
M M
M
U Y M M
M M
Y M M M
M M
M M M M
M M
M
M M
Therefore there are four arrangements stating with “y” AND “U” and twelve arrangements starting with “M” and altogether 12+4+4=20 different arrangements which have shown by drawing tree diagram.
2) Four letters the same.
“ABCCCC”
There are 30 arrangements for a word with six letters and for of them are same.
3) The word with five letters the same.
- ACCCCC
- CCCCCA
- CACCCC
- CCACCC
- CCCACC
- CCCCAC
For the word with six letters and five letters the same , there are six different arrangements.
Now I am going to draw a table and lok at the relationships, for the word with different number of letters the same.
Table 3
I have noticed that for the word with different number of (one letter the same) of letters the same: the number of different arrangements equal to the factorial of the total number of the number of letters the same.
Examples for MUMMY, there are five letters in the word and three letters the same “M, M, M” SO 5! /3!
Let “N”=number of letters in the word
“A”= number of different arrangements
R=number of letters the same.
Therefore N! /R! =A
Prediction:
Testing the formula
N! R! =A
e.g. =MUMMY
=5!/3!=120/6=20 different arrangements
This proves my formula is right and it would work for the word that has any number of letters the same.
Multiple number of letters the same
Now I am going to look at the group of letters that has more than one letters the same, example: “ABBCC, AABBCC,” etc
First of all I am going to look at “AABB”
- AABB
- ABAB
- BABA
- BBAA
- BAAB
- ABBA
There are six different arrangements.
I know numerator should be N!. I know this from the previous section.
From this section I have noticed that for the word with multiple number of letters the same..
The number of different arrangements is equal to the factorial of the different no of multiple letters the same.
Example: there are four letters in the word and two different the same
So 4! /2!*2!
Let “N”=number of letters in the word
“A”= number of different arrangements
R
Q =number of letters the same.
p different number of letters appears this number of time e.g. P times.
In a “N” letter word same letters appears “P, Q, R” times. So the number of ways we could arrange is
N! /P!*Q!*R! =A
Conclusion
In this investigation, I have learned how to find the different number of arrangements for the word with any of different letters or the word with different number of letters the same.
I found three different formulae:
- Word with different letters N! /A
- Words with more than one letter the same N! /R! =A
- Words with multiple letters the same N! /R!*P! =A