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  • Level: GCSE
  • Subject: Maths
  • Word count: 1590

Emma's Dilemma.

Extracts from this document...

Introduction

Emma’s Dilemma

Introduction

This coursework is about rearrangements of the letters in peoples names. We will be looking at how many combinations you can make out of the letters in different size names. I will then work out a formula that will let me work out the number of combinations in a name given the number of letters in that name. I will work this out in a systematic way, by changing one letter at a time and moving down the word.

The names I will be using are:

2 letter name

BO

3 letter name

SAM

4 letter name

MIKE

5 letter name

KARIN

Results

2 letter name

BO  OB

For this name, there are only 2 possible combinations

3 letter name

SAM  ASM  MSA

SMA  AMS  MAS

For the 3 letter name, there are 6 possible ways.

4 letter name

MIKE  IMKE  KMIE  EMIK

MIEK  IMEK  KMEI  EMKI

MKIE  IEMK  KIEM  EIKM

MKEI  IEKM  KIME  EIMK

MEIK  IKEM  KEIM  EKMI

MEKI  IKME  KEMI  EKIM

24 different combinations are possible with a 4 letter name

5 letter name

KARIN  KRAIN  KINAR  KNARI

KARNI  KRANI  KINRA  KNAIR

KAINR  KRNIA  KIARN  KNRAI

KAIRN  KRNAI  KIANR  KNRIA

KANRI  KRINA  KIRAN  KNIRA

KANIR  KRIAN  KIRNA  KNIAR

ARINK  AINKR  ANKRI  AKRIN

ARIKN  AINRK  ANKIR  AKRNI

ARKIN  AIKRN  ANIKR  AKRNI

ARKNI  AIKNR  ANIRK  AKINR

ARNKI  AIRKN  ANRKI  AKNIR

ARNIK  AIRNK  ANRIK  AKNRI

RINKA  RNKAI  RKAIN  RAKIN

RINAK  RNKIA  RKANI  RAKNI

RIKAN  RNAKI  RKINA  RANIK

RIKNA  RNAIK  RKIAN  RANKI

RIAKN  RNIAK  RKNIA  RAIKN

RIANK  RNIKA  RKNAI  RAINK

NIKAR  NKIAR  NAIRK  NRAKI

NIKRA  NKIRA  NAIKR  NRAIK

NIARK  NKARI  NARIK  NRKAI

...read more.

Middle

I will now work out a formula that will let me work out how many combinations there are in words with 2 letters the same. I will use the following names:

2 letter name

AA

3 letter name

AAL

4 letter name

ALAE

5 letter name

ANNIE

Also, I will look at the name Emma as part of our investigation, and to help prove my results and formula correct.

RESULTS

2 letter name

AA

For this, there is only 1 possible combination.

3 letter name

AAL  

LAA

ALA

3 possible combinations are possible with this name

4 letter name

ALAE  LAEA  EAAL

ALEA  LAAE  EALA

AAEL  LEAA  ELAA

AALE

AELA

AEAL

For a 4 letter name, there are 12 different combinations

5 letter name

ANNIE  ENNIA  INNAE  NNAEI  NEANI

ANNEI  ENNAI  INNEA  NNAIE  NEAIN

ANEIN  ENAIN  INENA  NNEAI  NEIAN

ANIEN  ENANI  INEAN  NNEIA  NEINA

ANENI  ENINA  INANE  NNIAE  NENAI

ANIEN  ENIAN  INAEN  NNIEA  NENIA

AENIN  EANNI  IENNA  NIEAN  NAINE

AEINN  EANIN  IENAN  NIENE  NAIEN

AENNI  EAINN  IEANN  NINAE  NAENI

AINEN  EINNA  IANNE  NINEA  NAEIN

AINNE  EIANN  IAENN  NIAEN  NANIE

AIENN  EINAN  IANEN  NIANE  NANEI

For a 5 letter name there are 60 possible combinations.

RESULTS

For names with 2 letters the same, I have done some examples, and am now going to show my results.

Table of Results

Number of letters the same

Letters in word

Combinations

2

2

1

2

3

3

2

4

12

2

5

60

Graph of Results

image01.png

FORMULA

I have now worked out a formula and will now explain it:

For a 2 letter word: 2x1=1

                                  2

For a 3 letter word: 3x2x1= 3

                                   2

For a 4 letter word: 4x3x2x1=12

                                      2

For a 5 letter word:

...read more.

Conclusion

RESULTS

Here are my results

Table of Results

Letters in Word

Words the same

Possible Combinations

3

3

1

4

3

4

5

3

20

Graph of Results

image02.png

FORMULA

I have looked at my results, and am now able to work out a formula.

For a 4 letter word: 4x3x2x1=4

                                  3x2x1

For a 5 letter word: 5x4x3x2x1= 20

                                   3x2x1

These results are correct, so therefore my formula is correct. I will now write out this formula in algebraic form.

C=N!/S!

C= Combinations

N= Number of letters

S = Same letters

! = Factorise

Predictions

Based on my formula, I will now make a few predictions

Letters in Word

Words the Same

Possible Combinations

6

3

120

7

3

840

8

3

6720

9

3

60480

10

3

604800

The reason we divide it by 3 factorised, is because there are 3 letters the same, and if we didn’t divide it, it would give the results as if each same letter was different. This means we must divide it by 3 factorised to get the right result.

Names with 4 letters the same

Now I will look at names with 4 letters the same. I will use the following names

4 letter name

AAAA

5 letter name

AAAAL

RESULTS

4 letter word

AAAA

Only 1 possible combination

5 letter word

AAAAL

LAAAA

ALAAA

AALAA

AAALA

5 possible combinations

RESULTS

Table of Results

Number of letters

Letters the same

Combinations

4

4

1

5

4

5

Graph of Results

image03.png

FORMULA

Now I will attempt to find  a formula based on my results

For a 5 letter word: 5x4x3x2x1=5

                                 4x3x2x1

This is correct, which proves that my results were correct. I shall now show the algebraic formula, which is

C=N!/S!

C= Combinations

N= Number of letters

S = Same letters

! = Factorise

PREDICTIONS

Number of Letters

Letters the Same

Combinations

6

4

30

7

4

210

8

4

1680

9

4

15120

10

4

151200

...read more.

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