There are 2 different letters in this name and there are 2 different arrangements.
Table of Results
- From the table of results I have found out that a 2-letters word with individual letters has 2 arrangements, and a 3-letter word with individual letters 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 that 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.
The formula for this is: n! = a
-
Where n = the number of letters in the word and
a = the number of different arrangements.
Now I am going to investigate the number of different arrangements in a word with 2 letters repeated, 3 letters repeated and 4 letters repeated.
- This is a 3-letter word with 2 letters repeated
MUM MMU UMM
3-letter word, 2 letters repeated, 3 different arrangements.
- This is a 5 letter word with 3 letters repeated
PQMMM PMQMM PMMQM PMMMQ
QPMMM QMPMM QMMPM QMMMP
MPQMM MPMQM MPMMQ MQPMM
MQMPM MQMMP MMPQM MMQMP
MMMPQ MMMQP MMPQM MMMQP
- 5-letter word, 3 letters repeated, 20 different arrangements.
- This a 4- letter word with 3 letters repeated
SMMM MSMM MMSM MMMS
4-letter word, 3 letters repeated, 4 different arrangements.
- This is a 5-letter word with 4 letters repeated
RRRRK RRRKR RRKRR RKRRR KRRRR
5-letter word, 4 letters repeated, 5 different arrangements.
Table of Results
I have worked out that if I say 5! = 120, to find out how many different arrangements in a 3 letter word (3x2 x1=6) it would be 5! Divided by 6= 20, so, a 6-letter word with 4 letters (4x3x2x1=24) repeated would be 6! Divided by 24 = 30, as you can see in the “No letters repeated” column these are the numbers we are dividing by:
(2x1 = 2)
( 3x2x1 = 6)
(4x3x2x1 = 24)
5 letters the same = n!
(5x4x3x2x1 = 120)
6 letters the same = n!
(6x5x4x3x2x1 = 720)
From this I have worked out the formula to find out the number of different arrangements:
n! = The number of letters in the word
p! = The number of letters the same
Now I am going to investigate the number of different arrangement for words with 2 or more letters the same like, aabb, aaabb, or bbbaaa.
“aabb”
aabb abba abab
baba bbaa baab
- This is a 4 letter word with 2 letters the same, there are 6 different arrangements:
I am going to use the letters x and y (any letter)
This is a 5-letter word
“Xxxyy”
xxxyy xxyxy xxyxx xyxyx xyxxy
xyyxx yyxxx yxxxy yxyxx yxxyx
There are 10 different arrangements 5-letters 3 the same and 2 different
In the above example there are 3 x's and 2 y's
As each letter has its own number of arrangements i.e. there were 6 beginning with x, and 4 beginning with y, I think that factorial has to be used again.
As before, the original formula:
n! = The number of letters in the word
p! = The number of letters the same
From this I have come up with a new formula. The number of total letters factorial, divided by the number of x's, y's etc factorised and multiplied.
For the above example:
A four letter word like aabb; this has 2 a's and 2 b's (2 x's and 2 y's)
So: 1x2x3x4 (Four letter word) = 24 = 6 different arrangements (1x2) x (1x2) (2a’s) x(2b’s) 4
A five letter word like aaaab; this has 4 a's and 1 b (4 x's and 1 y)
So: 1x2x3x4x5 (5 letter words) = 120 = 5 different arrangements
(1x2x3x4) x (1) (4a’s x 1b) 24
A five letter words like abcde; this has 1 of each letter (no letters the same)
So: 1x2x3x4 = 24 = 24 different arrangements
(1x1x1x1x1x1) 1
A five letter word like aaabb; this has 3 a's and 2 b's (3 x's and 2 y's).
So: 1x2x3x4x5 = 120 =10 different arrangements
(1x2x3) x (1x2) 12
This shows that my formula works:
n! = The number of letters in the word
x!y! = The number of repeated letters the same
Conclusion
I was able to find these formulas
-
The formula for this is: n! = a
Where n = the number of letters in the word and
a = the number of different arrangements
- This formula is used to find out the different arrangements and possibilities from individual letters in a word.
(2x1 = 2)
( 3x2x1 = 6)
(4x3x2x1 = 24)
(5x4x3x2x1 = 120)
(6x5x4x3x2x1 = 720)
- Finding similarities in letters the formula I found are shown above.
