# EMMA'S DILEMMA

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Introduction

### EMMA’S DILEMMA

#### INTRODUCTION: I will begin my Investigation by finding the number of arrangements does the word LUCY, EMMA and other words have. By doing this I will come up with a formula, which I could find, any number of arrangements, repeated words or no repeated words.

I will begin by finding the total arrangement does the word LUCY have. Then I will find the arrangement for a 1-lettered word, 2-letered word, 3-lettered word and then 5-lettered word, with no repeated letters. After that I will look at the arrangements and see if there is a pattern to the arrangements and find a formula, which can find any arrangements with no letter repeated.

##### Part 1

I will start my investigation by finding the number of arrangements does the word LUCY have.

- LUCY 7. ULCY 13. CULY 19. YLUC
- LUYC 8. ULYC 14. CUYL 20. YLCU
- LCUY 9. UYLC 15. CLUY 21. YULC
- LCYU 10. UYCL 16. CLYU 22. YUCL
- LYCU 11. UCLY 17. CYLU 23. YCLU
- LYUC 12. UCYL 18. CYUL 24. YCUL

I have found 24 arrangements in the word LUCY. Now I am going to find out how many number of arrangements does a three-letter word, two letter word and one letter word have.

###### 1-lettered word:

1.A

I have found only one arrangement in a one-letter word.

2-lettered word:

1.CA 2.AC

I have found only two arrangements in a two-lettered word.

3-lettered word:

1. CAN 3. ACN 5. NAC

2. CNA 4. ANC 6. NCA

I have found only six arrangements in a three-lettered word.

Middle

Lets see what’s wrong with my formula.

2 letters are same, then:

3 letters: 1*2*3/1*2

4 letters: 1*2*3*4/1*2

5 letters: 1*2*3*4*5/1*2

So to get the arrangement of a word with two same letters, then you have to divide it by 2. Lets see how you get the arrangement of a word with 3 letters same.

3 letters are same, then:

3 letters: 1*2*3/1*2*3

4 letters: 1*2*3*4/1*2*3

5 letters: 1*2*3*4*5/1*2*3

And so on.

I have realised that, to find the arrangement of a 4 lettered word with 3 same letters.

I have to get the arrangement of the 4-lettered word, which is 24. Then I have to find the arrangement of the repeated letters. So there are 3 letters same in a 4 lettered word. So the arrangement of the repeated letter is 6. Because the arrangement of a 3 lettered word is 6.

Then you have to divide 24 by 6 and you will get the answer, which is 4.

My prediction is that to find the arrangement of a word with repeated letters, you have to first get the arrangement of the word. Then you have get the number of arrangements of how many repeated letters there are. Then you divide both the number of arrangement of the word and the number of arrangement of the repeated letters. Then you will get the arrangement. This is my prediction.

I have given a table below to show my prediction. I will call the prediction table, table 3

Prediction table:

3 letters same:

Table 3

Number of letters | Arrangement |

3 | 1 |

4 | 4 |

5 | 20 |

6 | 120 |

Conclusion

So the formula is:

A=N! / R!

A= Number of arrangements

N= Number of letters

R= Number of repeated letters

!= Number of arrangements for (N-1)

This formula can also be done in a calculator. By typing in the number of letters and then pressing '!’. You should get the arrangement.

These letters I put are important. Without these letters you cannot find the arrangement. The letter ‘A’ tells you what the arrangement is. Without that letter you would not know the arrangement. So the ‘A’ letter is important. The letter ‘N’ tells you how many letters are in the word. With this letter you need to find out the arrangement of a no repeated word. So this letter is important. The Letter ‘R’ is the final letter in the formula. This letter tells you how many repeated letters there are. Without this letter you cannot find the arrangement of a word with repeated letters in it. Last of all is the icon ‘!’. This icon is very important. This icon is used for everything. To find an arrangement for a repeated letter or no repeated letter. This icon cannot be removed from the formula because without that icon you cannot find the formula.

EVALUATION

From this investigation I have learned how to find arrangements and work out the formula. This investigation was quiet exciting but it was also quite tiring because of finding the arrangement. This investigation was long and quite hard as well. It was quite challenging investigation.

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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