Investigating the arrangements of letters in words.

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Introduction

I am doing an investigation into words and the number of ways the words can be written.  I will find formulae and work out the number of combinations for the words.  

Originally the task was to find combinations for the word MATT, but I decided to look at sequences of letters from the alphabet, which makes it easier to monitor and control because letters can be written in sequence eg. A–B–C.

I will start by looking at words with no letters the same because this is the simplest to work with. I will find a formula for that, beginning with words that are 1 letter long and carrying on to letters that are 5 letters long so I have a wide range of data to form a formula out of and test it .

        E.g.

                ABCD

Next I will look at words that have two letters the same e.g. AABC because the name Matt has 2 T’s in it. I will be using the same method as before but starting with 2 letter words and expanding the amount of letters to 5 or 6 letter words so again I will have a significantly large range of data to for a formula from. By doing this I am expanding the investigation and gaining more knowledge of the pattern of formulae.

Throughout the investigation I will use 3 algebraic expressions. 

n =number of letters in total within the word.

c= number of combinations found in total

n!=n factorial

Also I will be using simplified diagrams for large numbers.

eg

This works because the underlined bit (ABC) can be written in 6 different ways shown below, and the addition of the D means that the letters can be written around the D in 6 different ways and the D moved 4 times to a different position so the ABC can be written around the D 4 different ways so the total number of arrangements is    6 x 4 which = 24

Investigation

I am going to investigate the different ways in which letters can be organised and find formulas for all of them.

Letters that are all different

I will now try and find a formula for this pattern.  I will put the data in a table to make it clearer first:

I will now prove that 5 different letters has 120 combinations:

ABCDE=24

ABCED=24

ABECD=24

AEBCD=24

EABCD=24

Total=120

I worked out the formula was n! after first discovering the pattern was 1x2; 1x2x3; as so on.  From prior knowledge I knew an easier way to write this was n! or n factorial. The factorial pattern is:

1!=0x1=1

    2!=0x1x2=2

        3!=0x1x2x3=6

              4!=0x1x2x3x4=24

I can say the formulae for words where each letter is different (or 1 letter is the same) is:

n!

Proof

The formula is n! because it is a simpler way or writing N x N-1 x N-2…till N-x = 1. The formula is N x N-1 x N-2…till N-x = 1 because when a new letter is added the previous number (N-1 ) of arrangements can be written around it N times and to find N-1 you need to find the previous number of arrangements again and again till you get to 1. N! simply means N x N-1 x N-2…till N-x = 1.


2 Letters the same

I will now try and find a formula for this pattern by putting all the data in a table to make it easier to understand and observe:

Join now!

Now, I’ll prove that there are 120 combinations in a 5 letter word:

ABBCD=24

ABBDC=24

ABDBC=24

ADBBC=24

DABBC=24

Total= 120

I worked out the formula was n!/2 because I put the figures for n! alongside the combination which showed me the combination was exactly half that of the n! so it seemed logical to test n!/2 which worked.

n!/2

        

Proof

Looking at this the least number of combinations 1 is achieved buy a 2 letter word. This means that the number combination goes 1 3 12 when before it was 1 2 6 24, by ...

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