# Emma&#146;s Dilemma

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Introduction

Emma’s Dilemma

In this investigation I will try and work out how many combinations I can make out of any type of word.

4 letters

The name EMMA has 4 letters, with 2 letters the same. I will experiment how many combinations I can make with this name.

I find it easier to try and make each combination using one letter, then another etc.

EMMA MEMA MEAM AMME

EMAM MMEA MAEM AMEM

EAMM MMAE AEMM

The name LUCY has 4 letters, which are all different.

LUCY LYUC ULCY CULY CUYL YLUC

LCUY UCLY UYLC CYUL YCLU YULC

LYUC UCYL UYCL CLYU YLCU YUCL

LCYU ULCY CYLU CLYU YCUL LUYC

The name SASS has 4 letters, 3 of which are the same.

SASS SSSA

SSAS ASSS

The name AAAA has 4 letters, which are all the same.

AAAA has no variations.

3 letters

The name LOU has three letters, all different.

LOU ULO OUL

Middle

No. of letters | All different | 2 the same | 3 the same | 4 the same | 5 the same |

2 | 2 | 1 | |||

3 | 6 | 3 | 1 | ||

4 | 24 | 12 | 4 | 1 | |

5 | 5X24= 120 | 120 2=60 | 60 3=20 | 20 4=5 | 1 |

From these results I have now found out how this works. The number of combinations is narrowed down each time there is more that 1 letter the same, or 1 letter is chosen to begin.

This is what I mean:

E.g. 2 letter word

2 choices for letter, multiplied by one choice for letter = 2

3 letter word

1

Conclusion

E.g. 10 ! = 10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1

10! = 3, 628, 800

I can therefore say that a 10 letter word with all of the letters different could make 1, 628, 800 combinations of itself.

I have come up for this equation for finding the combination of an n letter word, with all letters different.

N! = n x (n - 1) x (n - 2) x (n - 3) x (n - 4) x (n - 5) x (n - 6) x (n - 7) … etc.

If 2 letters were the same, the equation would be:

n! 2

And if 3 letters were the same, it would be:

n! 2 3 …etc.

This concludes my coursework, as I have now found how many combinations you can get from an n letter word, even when the word has 2 letters the same, 3 letters the same, 4, or 5 …etc.

- Louise Manly

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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