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Emma's Dilemma

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Introduction

Mathematics GCSE

EMMA’S DILEMMA

1.

No. of arrangements

Arrangements

1

EMMA

2

MEAM

3

AEMM

4

MMEA

5

MMAE

6

AMEM

7

EMAM

8

AEMM

9

MAEM

10

AMME

11

MAME

12

MEMA

 2.

No. of arrangements

Arrangements

1

LUCY

2

LYCU

3

LYUC

4

LUYC

5

LCYU

6

LCUY

7

UCYL

8

UYCL

9

ULCY

10

ULYC

11

UYLC

12

...read more.

Middle

YLCU

21

YUCL

22

YULC

23

YCLU

24

YCUL

   3.

No. of arrangements

No. of letters

Position(n)

1

A

1

2

AB

2

6

ABC

3

24

ABCD

4

120

ABCDE

5

720

ABCDEF

6

5040

ABCDEFG

7

40320

ABCDEFGH

8

362880

ABCDEFGHI

9

3628800

ABCDEFGHIJ

10

I have completed the table by using a method I found after finding a pattern in the first few results. For a four-letter word (all different letters) there are 24 arrangements as I found in Question 2 (LUCY). There are six combinations beginning with each letter, with this knowledge I think that a five-letter word (all different letters) would have 120 arrangements. To prove this answer/back it up I have found another pattern in the table.

4 letter word         1x2x3x4=24 (No. of arrangements)

...read more.

Conclusion

3

2

Aabb

6

3

Aaabb

10

4

Aaaabb

15

5

Aaaaabb

21

6

Aaaaaabb

28

7

Aaaaaaabb

36

8

aaaaaaaabb

45

To further my investigation I have decided to investigate words with 3 different letters. First I will try to find the number of arrangements for the word abc using the formula then I will check it is correct by working it out myself, if the 2 results are the same then I will carry on finding the number of arrangements for words with 3 different letters in and produce a table.

3 letter word / 1a  x  1b  x  1c  = No. of arrangements

    1x2x3        /1x1 x 1x1 x 1x1 = 6/1 = 6

  1. abc
  2. acb
  3. bca
  4. bac
  5. cab
  6. cba

Both methods produced the same number of arrangements (6).

Position (n)

Words arranged

No. of arrangements

1

Abc

6

2

Aabc

12

3

Aaabc

20

4

Aaaabc

30

5

Aaaaabc

42

6

Aaaaaabc

56

7

Aaaaaaabc

72

8

aaaaaaaabc

90

...read more.

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