To investigate the number of different arrangements of letters in a different words

This means that the formula for the number of arrangements for a word with no repeated letters is:

n! = A

When n is the number of letters and A is the number of arrangements

I tested this formula and proved it worked. My evidence follows:

LUCY – 4 letters all the same

n = 4

4! = 4x3x2x1 = 24

Using this I predicted the following results

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.

I am going to start with EMMA

EMMA

AMME

AMEM

EMAM

AEMM

EAMM

MMEA

MMAE

MEMA

MAME

MEAM

MAEM

 

4-letter word, 2 letters repeated, 12 different arrangements.

MUM

MMU

UMM

 

3-letter word, 2 letters repeated, 3 different arrangements.

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.

 

SMMM

MSMM

MMSM

MMMS

 

4 letter word, 3 letters repeated, 4 different arrangements.

 

RRRRK

RRRKR

RRKRR

RKRRR

KRRRR

 

5 letter word, 4 letters repeated, 5 different arrangements.

When looking at the number of arrangements for Emma I realized that there were 12 and this was half of the number of arrangements for Lucy so I decided to look at the others. This also was the case for the 3 letter words such as Mum and Sam.

24/2 = 12 – The number of arrangements for Emma.

6/2 = 3 – The number of arrangements for Mum.

I started to look at what they had in common apart from the divide by two. I noticed that they both had 2 letters repeated. I decided to check other arrangements to see if with 2 letters you had to divide the number of arrangement if there were no repeats by 2.

I looked at a word with five letters and 2 repeated letter I came up with:

Small. There are 60 arrangements which is half of 120 which is the number of arrangement for a 5 letter word with all the letters the same. So it was obvious that for a word with 2 letters repeated there it was always divided by 2.

I began to ask why it was 2 and came up with the idea of checking to see what number if any you have to divide by to get the number of arrangements for different amounts of letter arrangements.

I found that for the letter arrangement AAASD there were 20. Going on what I had found out before it became clear that for a word or arrangement with 3 letters repeated you had to divide by 6.

I looked into it and discovered that both 2 and 6 were the factorial numbers of 2 and 3. Which coincided with the number of letters repeated. I came up with the formula to work out the number arrangements for any word with 1 letter repeated any number of times:

n!/r! = a

When n = the number of letters in the word, r = number of times the letter is repeated and a = number of arrangements.

This is because if you had a word for example:

EMMA there are 12 different arrangements, this is only because the Ms and repeated so for example if the on arrangement was EMMA  and another EMMA although when you use two different colours it counts as two arrangements but when colour is taken away and you are left with the raw word you only end up with half the number of possible arrangements (only in this case though as it depends on the number of letters repeated).

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.

 

This is a 4 letter word with 2 letters the same, there are 6 different arrangements:

xxyy

 

I am going to use the letters x and y (any letter)

 

xxyy     xyxy     yxxy

xyyx     yxyx     yyxx

 

This is a 5 letter word

xxxyy

 

xxxyy   xxyxy   xxyxx   xyxyx   xyxxy

xyyxx   yyxxx   yxxxy   yxyxx   yxxyx

 

There are 10 different arrangements

In the above example there are 3 xs and 2 ys

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

       (1x2) x (1x2)

= 24

    4      = 6 different arrangements

 

A five letter word like aaaab; this has 4 a's and 1 b (4 x's and 1 y)

So: 1x2x3x4x5

      (1x2x3x4) x (1)

           = 120

               24     = 5 different arrangements

 

A five letter words like abcde; this has 1 of each letter (no letters the same)

So : 1x2x3x4

   (1x1x1x1x1x1)

          = 24

              1        = 24 different arrangements

 

 

A five letter word like aaabb; this has 3 a's and 2 b's (3 x's and 2 y's).

So : 1x2x3x4x5

       (1x2x3) x (1x2)

= 120

    12    = 10 different arrangements

 

 

 

This shows that my formula works:

                       

          n!   = the number of letters in the word

        x!y! = the number of repeated letters the same