# Emma's Dilemma

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Introduction

Maths Coursework – Emma’s Dilemma

Emma’s Dilemma

Plan

I am investigating the number of different arrangements there will be in different types of names. Some names I will investigate on will have no identical letters such as LUCY. Some will have a pair of identical letters such as EMMA. Some names will have different quantities of letters such as AMMIE, JOE and ANNE.

Firstly, I’ll produce a method which will help me figure out the different arrangements in the name EMMA and LUCY without using any formulas. Using this method I’ll test all my predictions.

I’ll create formulas for different types of names. Examples are names with no identical letters, a pair or more identical letters and names that have 2 different groups of identical letters

Throughout the investigation I will devise different formulas for working out different types of names. By using one of these formulas I’ll figure out the total number of arrangements of the letters XX……XXYY…….Y.

To start of the investigation I have to find the different arrangements of letters in the name EMMA and LUCY without the use of any types of formulas. Therefore, I have developed my own method.

Method

- Write the name in its original format. In this case it will be EMMA or LUCY.

Middle

JOE= 3X(2X1)= 6

JOHN= 4X(3X2X1) = 24

By looking at this pattern the arrangements of a 5-letter word will be = 5X(4X3X2X1) = 120

A 6 letter word with no identical letters would equal = 6X(5X4X3X2X1) = 720

These type of sequences are called number FACTORIALS, which are represented by the ! Button on the scientific calculator.

So the formula for calculating the number of different arrangements in a word with no identical letters is:

A=N!

A = total number of arrangements

N = the number of letters in the word.

Testing and predictions

To test if the formula N! is correct I will use it to figure out the number of total arrangements in the name JOE.

Prediction:

There are 3 letters in the name JOE.

So 3! = 3X2X1 = 6

Testing

Starting letter J | Starting letter O | Starting letter E |

JOE JEO | OJE OEJ | EJO EOJ |

Overall Total: 6 |

My Prediction is correct!

What if a name had a pair of identical letters

From the name EMMA, I already know a 4 letter word with a pair (2 identical letters) of identical letters can be arranged 12 times.

To make this theory correct I will investigate the number of arrangements there will be in the name ANNE.

Starting letter “A” | Starting letter “N” | Starting letter “E” |

ANNE ANEN AENN | NANE NAEN NEAN NENA NNEA NNAE | EANN ENAN ENNA |

Total: 3 | Total : 6 | Total: 3 |

Overall total: 12 |

Arrangements in AMMIE’s name

Starting Letter “A” | Starting Letter “M” | Starting Letter “I” | Starting Letter “E” |

AMMIE AMMEI AMIME AMIEM AMEMI AMEIM AIMME AIMEM AIEMM AEMMI AEMIM AEIMM | MAMIE MAMEI MAIME MAIEM MAEMI MAEMA MMAIE MMAEI MMIAE MMIEA MMEAI MMEIA MIAME MIAEM MIMAE MIMEA MIEMA MIEAM MEAMI MEAIM MEMIA MEMAI MEIAM MEIMA | IAMME IAMEM IAEMM IMAME IMAEM IMMAE IMMEA IMEAM IMEMA IEAMM IEMAM IEMMA | EMMAI EMMIA EMAMI EMAIM EMIMA EMIAM EAMMI EAMIM EAIMM EIAMM EIMAM EIMMA |

Total: 12 | Total: 24 | Total: 12 | Total: 12 |

Overall Total: 12X3+24= 60 |

Conclusion

There are many ambiguous answers I got through my investigation.

Firstly, I tried the formula; A=N!/X! on the name Lucy. Since there wasn’t any identical letters in the name, I substituted X for 0 which gave me the whole answer of 0. This gave me the idea that the A=N!/X! only suitable for words with 2 identical letters.

Secondly, I tried the same formula for the rows; XX……XXYY…….Y, I got the equation of 20!/10!, which gave me the ambiguous answer of 670442572800. This made me realise I was wrong so I created a new formula to help me figure this row out.

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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