How does Nike benefit from competitive advantage?

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Introduction 

In my investigation I am going to look at the number of different arrangements of letters in a word and the formulae that connect them. I am going to start by finding the formula for the number of arrangements for words which have all different letters. I am then going to continue by finding the formula for the number of arrangements for words with two letters the same. This will be carried on to find the number for different arrangements of various groups of letters and a formula will be determined that connects them all.

Part 1

Aim

To look at words with no duplicated letters, finally investigating the number of different arrangements of the letters of Lucy’s name.

With 1 letter in the word there is 1 arrangement.

1) K

This is because there is only 1 space for the letter to go.

1

With 2 letters in the word there are 2 different arrangements.

1) JO 2) OJ

This is because there are 2 choices for the first letter and 1 for the last.

2 1

 

With 3 letters in the word there are 6 different arrangements.

1) LIZ 2) LZI 3) ILZ 4) IZL 5) ZLI 6) ZIL

This is because there are 3 choices for the first letter, 2 for the next and then 1

3 2 1

 

 

 

From this initial assessment the rule for any size word combination, where all the letters are different, is a= n!  where n is the number of letters in the word and a is the total number of arrangements.

I would predict, therefore, that there will be 4! or 24 arrangements in the word Lucy. My check is below:-  

  1)     LUCY           7) ULCY              13) CULY             19) YLUC

  2)     LUYC           8) ULYC              14) CUYL             20) YLCU

  3)     LYCU           9) UCYL              15) CYUL              21) YUCL

  4)     LYUC           10) UCLY        16) CYLU              22) YULC

  5)     LCYU            11) UYLC        17) CLUY              23) YCUL

  6)     LCUY           12) UYCL        18) CLYU               24) YCLU

This confirms my prediction and the basis for the formula   a=n!

This can also be reached by multiplying the numbers out in Lucy.

4x3x2x1

 

I came to this prediction because in a four-lettered word there are 4 choices for the first letter

4---

For the second letter there are then 3 choices

4 x 3--        

for the next there are 2 choices                        

4 x 3 x 2-

and for the last, there is only 1 choice

4 x 3 x 2 x 1

this theory works for any number of letters.

Part 2

Aim

a) My investigation will now continue by assessing the number of different arrangements of a word with two letters the same, finally investigating the number of different arrangements of the letters of Emma’s name.

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Given a 2-letter word, with 2 letters repeated, there is 1 arrangement.

1) AA

Given a 3-letter word, with 2 letters repeated, there are 3 arrangements.

1) ANN 2) NAN 3) NNA

Given a 4-letter word, with 2 letters repeated, there are 12 arrangements.

1) EMMA 2) AMME 3) AMEM 4) EMAM 5) AEMM 6) EAMM.7)  MMEA 8) MMAE 9) MEMA 10) MAME 11) MEAM 12) MAEM

 

 

The connection between the two sets of data is that the words with two letters the same, have half the number of arrangements that the words with all different letters have.

 

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