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• Level: GCSE
• Subject: Maths
• Word count: 1424

# Investigate the number of different arrangements of the letters in a name.

Extracts from this document...

Introduction

Page 1

Emma’s Dilemma

Emma is playing around with arrangements of the letters of her name.

One arrangement is EMMA, a different arrangement is MEAM. Another arrangement is AEMM.

1. Investigate the number of different arrangements of the letters of Emma’s name.

1. EMMA

2. AEMM

3. MAEM

4. MMAE

5. MMEA

6. AMME        12 WAYS

7. EAMM

8. MEAM

9. MEMA

10. AMEM

11. MAME

12. EMAM

Emma has a friend called Lucy.

1. Investigate the number of different arrangements of the letters of Lucy’s name.

1. LUCY

2.YLUC

3. CYLU

4. UCYL

5. UCLY

6. YUCL

7. LYUC

8. CLYU

9. CLUY

10. YCLU

11. UYCL

12. LUYC

13. ULYC

14. CULY

15. YCUL

Page 2

16. LYCU      24 WAYS

17. YLCU

18. UYLC

19. CUYL

20. LCUY

21. LCYU

22. ULCY

23. YULC

24. CYUL

Choose some different names.

1. Investigate the number of different arrangements of the letters of the names you have chosen.

These names all have all letters different:

1.Jo

2.Max

3.Mark

4.James

1. JO    2 WAYS
2. OJ
1. MAX
2. XMA

Middle

MSAJEEMSAJJEMSAAJEMSAJESMMAJESSMAJEESMAJJESMAJSEMAAJSEMMAJSEEMAJSSEMAJSEMJAASEMJJASEMMJASEEMJASMEJASSMEJAASMEJJASMEEJASMJMASEEJMASSEJMAASEJMMASEJMASJEEMASJJEMASSJEMAASJEMASJMEEASJMMEASJ

Page 6

1. JMEAS
2. SJMEA
3. SJMAE
4. ESJMA
5. AESJM
6. MAESJ
7. JMAES
8. JAMSE
9. EJAMS
10. SEJAM
11. MSEJA
12. AMSEJ

120.          MAJES

These names all have two letters the same:

1.JJ

2.BOB

3.SAMM

4. SARAH

1. JJ         1 WAY
1. BOB
2. BBO    3 WAYS
3. OBB
1. SAMM
2. MSAM
3. MMSA
4. AMMS
5. AMSM
6. MAMS       12 WAYS
7. SMAM
8. MSMA
9. SMMA
10. ASMM
11. MASM
12. MMAS

Page 7

1. SARAH
2. HSARA
3. AHSAR
4. RAHSA
5. ARAHS
6. ARASH
7. HARAS
8. SHARA
9. ASHAR
10. RASHA
11. RASAH
12. HRASA
13. AHRAS
14. SAHRA
15. ASAHR
16. ASARH
17. HASAR
18. RHASA
19. ARHAS
20. SARHA
21. ASRHA
22. AASRH
23. HAASR
24. RHAAS
25. SRHAA
26. RSHAA
27. ARSHA
28. AARSH
29. HAARS
30. SHAAR
31. SHARA
32. ASHAR
33. RASHA
34. ARASH
35. HARAS
36. HARSA
37. AHARS
38. SAHAR
39. RSAHA
40. ARSAH
41. ASRHA
42. AASRH

Page 8

1. HAASR
2. RHAAS
3. SRHAA
4. SHRAA   6O WAYS
5. ASHRA
6. AASHR
7. RAASH
8. HRAAS
9. HRAAS
10. SHRAA
11. ASHRA
12. AASHR
13. RAASH
14. RAAHS
15. SRAAH
16. HSRAA
17. AHSRA
18. AAHSR

These names all have three letters the same:

1.

2.

Conclusion

6

Using this method I am able to predict the amount of different ways there are for names that haven’t got all letters the same. For example if I wanted to find the number of ways for a six letter word with two letters the same I would do 6!

2!

So it would be 6 (6-1) (6-2) (6-3) (6-4) (6-5)  which is 6 x 5 x 4 x 3 x 2 x 1

2 (2-1)                                               2 x 1

So it would be 720. The number of ways for a six letter word with two letters the same would be 360. Another prediction would be if N = 7 and D = 3 I would place them in the formula so it would be 7!  Which is 7 (7-1) (7-2) (7-3) (7-4) (7-5) (7-6)

3!                               (3-1) (3-2)

So it would be 5040  so the number of ways for a seven letter word with three

6

letters the same would be 840.

In conclusion using the formulas N! and N! make it easier to find the number of different ways of arrangements in a word.

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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