# I will find all the different combinations of Emma's name by rearranging the letters. Following this, I will do the same with Lucy's name and compare

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Introduction

By Emily Kho – 10GGW

Maths Investigation: Emma’s Dilemma

Introduction: During my investigation, I will explore the number of arrangements of different names. My aim is to find a general formula which will enable me to work out the number of arrangements for any name. I will begin this investigation with the name EMMA.

Method: I will find all the different combinations of Emma’s name by rearranging the letters. Following this, I will do the same with Lucy’s name and compare my results. All the letters in Lucy’s name are different so, I will investigate the number of permutations for names with different letters. From my results, I will devise a formula for names with different letters. Emma’s name has two letters which are the same so I will investigate the numbers of alternative combinations for names where two letters are the same. I will do the same for names where three letters are the same. I will compare these with my original results and work out a formula for names with different letters. My final investigation will be with names with sets of same letters e.

Middle

= 3x2x1

Lucy

4

24

= 4x3x2x1

From my results, I have found a pattern. In Lucy’s name, there are 4 letters and 24 possible arrangements. 24 is equal to 4x3x2x1. Therefore, in order to work out the number of arrangements for a name with different letters, you must multiply the number of letters by every whole number between itself and 1 inclusive.If the number of letters was n, this would be the formula:

n(n-1)(n-2)(n-3)…x3x2x1

This form of multiplication is called factorial. The symbol for factorial is ! Therefore, the formula is:

n!

This means that the number of arrangements for the name Emily is 5! which equals 120.

Next, I will investigate names where two letters are the same.

MM | MMA | MMAB | MMABC |

MM | MMA | MMAB | ??? |

MAM | MMBA | ||

AMM | MABM | ||

MAMB | |||

MBAM | |||

MBMA | |||

ABMM | |||

AMBM | |||

AMMB | |||

BAMM | |||

BMAM | |||

BMMA | |||

Here are my results:

Name | # Of Letters | # Of Arrangements |

MM | 2 | 1 |

MMA | 3 | 3 |

MMAB | 4 | 6 |

Compared to my previous results, these results are exactly half. This means that in order to find the number of arrangements for names with 2 same letters, you must divide our original formula, n! by 2. Therefore the formula for names where 2 letters are the same is:

n!

2

2 is equal to 2x1 so, the formula is:

n!

2!

This means that the number of combinations for the name, MMABC is 5! divided by 2! which equals 60.

I will now investigate names where 3 letters are the same.

MMM | MMMA | MMMAB |

MMM | MMMA | ??? |

MAMM | ||

AMMM | ||

MMAM |

Conclusion

MMLL | MMMLL |

MMLL | MMMLL |

MLML | MMLLM |

MLLM | MMLML |

LLMM | MLLMM |

LMML | MLMLM |

LMLM | MLMML |

LMMML | |

LMLMM | |

LLMMM | |

LMMLM | |

There are 6 combinations for the name MMLL and 10 for MMMLL. I have discovered that if we divide the number of letters factorial by the number of the 1st set of same letters multiplied by the number of the 2nd set of same letters, it will give us the number of arrangements. If the number of Ms = y and the number of Ls = x, this would be the formula for working out the number of combinations for names with sets of same letters.

n!

y!x!

This formula will work for any amount of sets of same letters. For example, the name Hannah which has 6 letters and 3 sets of 2 same letters. The answer would be:

6!

2!2!2!

The answer is 90.

Conclusion: During this investigation, I found a general formula to work out the permutations of names.

Names with different letters when n equals number of letters:

n!

Names with same letters when n equals number of letters and s equals the number of same letters:

n!

s!

Names with sets of same letters when n equals number of letters:

e.g. ANNA

N= a and A=b

n!

a! b!

These formulas that I have found will enable me to work out the number of permutations for any name.

Written by Emily Kho

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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