# Emmas dilemma

Extracts from this document...

Introduction

Investigation

I am first going to write out all the combinations for the name Emma.

emma

eamm

emam

meam

maem

mmea

mmae

mema

mame

amme

aemm

amem

This proves that if a name has the same arrangements of letters they will have the same total number of different combinations. However, this does not give me sufficient information to develop a formula. I am now going to look at names with 2 letters the same like Emma, but with a different total number of letters.

I am going to look at all the combinations for the name Lucy.

Lucy has 24 different combinations. This means that all names with 4 different letters will have a total of 24 different combinations. I am now going to look at different names with different numbers of letters.

Middle

resio

resoi

reiso

reios

reois

reosi

This means that I can work out the total number of combination by factorial notation.

I realised this because if I could find the total number of combinations by multiplying the total number of letters by the previous number of combinations it was the same as multiplying the total number of letters by its previous consecutive numbers Factorial Notation.

All letters Different | |||

Name | Number of Letters | Number of Combinations | |

Jo | 2 | 1 x 2 = 2 | 2 = 2 x 1 |

Ian | 3 | 2 x3 = 6 | 6 = 3 x 2 x 1 |

Lucy | 4 | 6 x 4 = 24 | 24 = 4 x 3 x 2 x 1 |

Rosie | 5 | 24 x 5 = 120 | 120 = 5 x 4 x 3 x 2 x 1 |

This means when a name has all letters different with any amount of total letters, I can work out the total number of combinations by using this formula.

Number of combinations = Total number of letters Factorial

C = t !

Ian

Total number of letters = 3

C = 3 !

3 ! = 3 x 2 x 1

3 ! = 6

C = 6

Conclusion

Name | Total number of letters | Total number of letters factorial (B) | Total combinations (A) | Difference between A and B |

Nn | 2 | 2 | 1 | 1 |

Ann | 3 | 6 | 3 | 3 |

Emma | 4 | 24 | 12 | 12 |

Jenny | 5 | 120 | ??? | ??? |

The difference between the total number of combinations and the total number of letters factorial is always total number of letters factorial divided by 2.

Therefore the formulae for the total number of combinations when 2 letters are the same is:

Combinations = total number of letters factorial / 2

C = t ! / 2

For Emma

C = 4 ! / 2

C = 12

This means I can predict the total amount of combinations for the name Jenny.

Combinations = total number of letters factorial / 2

Combinations = 5 ! / 2

Combinations = 60

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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