# Emma's Dilemma

Extracts from this document...

Introduction

Bhavni Shah

Maths Coursework

## Introduction

Emma’s Dilemma is about how many times a name or group of letters can be re-arranged.

Firstly I will start by re-arranging the name ‘Lucy.’ I will state all the possible ways of re-arranging ‘Lucy.’ I will also find a formula for it.

Secondly I will re-arrange the name ‘Emma’ and state the different re-arrangements. I will also find a formula for it.

I will then use a different set of letters and names and find out how many times they can be re-arranged using a formula I create for it.

Lastly, I will make my coursework harder by using words, which have more than one of the same letters, e.g. Mississippi. I will re-arrange AAB and then move on to more complicated ones like AABBCCDDEE.

I will find a formula that relates to them as well.

## Hypothesis

I think that the more letters there are, the more re-arrangements there would be whether there is more than one of the same letters. For e.g.

Middle

2

2

3

6

4

24

5

120

6

720

I have found out that when multiplying the group of letters by the previous number of arrangements, I get the number of arrangements of the one I require.

1 2 3 4 5 6

1 2 6 24 120 720

=equals

=formula

I have found out that when multiplying the group of letters by the previous number of arrangements, I get the number of arrangements of the one I require.

For e.g. if I want to find the number of re-arrangements for 7 letters, I will use the formula below.

Number of letters*previous number of re-arrangements

7*720=5040

- I will now show you the different ways in which ‘Emma’ can be written as.

EMMA MMEA AEMM

EAMM MMAE AMEM

EMAM MEAM AMME

MAEM

MEMA

MAME

There are 12 ways in which I can re-arrange the name ‘Emma.’ This is because there are 2 M’s. They may be the same letters but they are counted separately.

Finding the formula for ‘EMMA’ will be somewhat harder since it has ‘2 M’s.’ Firstly I will re-arrange groups of letters with 2 of the same letters and see if there is a link between them and ‘EMMA.’

Conclusion

3 T’s

2 M’s

3 N’s

This is what my calculator display looked like:

28!

(5!*4!*4!*2!*2!*2!*2!)

My answer equals to: 3

This is because there are so many letters that are repeats.

I also thought of re-arranging the name ‘MISSISSIPPI’ seeing as it has many repeated letters in it.

### MISSISSIPPI

There are:

4 I’s

4 S’s

2 P’s

I have found out that I can re-arrange the word ‘MISSISSPPI’ 34650 times. This is mainly because there are so many repeat letters.

### Conclusion

I have found out that my hypothesis was incorrect. I said that ‘the more letters there are, the more re-arrangements there would be whether there is more than one of the same letters.’

This statement is however incorrect as I soon discovered when finding out how many times I could re-arrange the name ‘EMMA.’

I realised that even though the number of letter were the same, there were repeats so the amount of times they could be re-arranged would not be the same.

To conclude, I would like to point out that the number of letters in a word is not the only thing that determines the number of arrangements possible but also repeats, if there are any.

- -

This student written piece of work is one of many that can be found in our GCSE Emma's Dilemma section.

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