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

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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.

- 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.

- 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.

- 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

- JO 2 WAYS
- OJ

- MAX
- XMA

Middle

Page 6

- JMEAS
- SJMEA
- SJMAE
- ESJMA
- AESJM
- MAESJ
- JMAES
- JAMSE
- EJAMS
- SEJAM
- MSEJA
- AMSEJ

120. MAJES

These names all have two letters the same:

1.JJ

2.BOB

3.SAMM

4. SARAH

- JJ 1 WAY

- BOB
- BBO 3 WAYS
- OBB

- SAMM
- MSAM
- MMSA
- AMMS
- AMSM
- MAMS 12 WAYS
- SMAM
- MSMA
- SMMA
- ASMM
- MASM
- MMAS

Page 7

- SARAH
- HSARA
- AHSAR
- RAHSA
- ARAHS
- ARASH
- HARAS
- SHARA
- ASHAR
- RASHA
- RASAH
- HRASA
- AHRAS
- SAHRA
- ASAHR
- ASARH
- HASAR
- RHASA
- ARHAS
- SARHA
- ASRHA
- AASRH
- HAASR
- RHAAS
- SRHAA
- RSHAA
- ARSHA
- AARSH
- HAARS
- SHAAR
- SHARA
- ASHAR
- RASHA
- ARASH
- HARAS
- HARSA
- AHARS
- SAHAR
- RSAHA
- ARSAH
- ASRHA
- AASRH

Page 8

- HAASR
- RHAAS
- SRHAA
- SHRAA 6O WAYS
- ASHRA
- AASHR
- RAASH
- HRAAS
- HRAAS
- SHRAA
- ASHRA
- AASHR
- RAASH
- RAAHS
- SRAAH
- HSRAA
- AHSRA
- 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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