Emma's dilemma

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Emma’s dilemma

THAMILINI KUGARAJAH

CATERHAM HIGH SCHOOL

CENTER NO: 13321

CANDIDATE NO: 9191

Introduction:

This investigation is about finding the different arrangements of letters in various words. I am going to look at the number of different arrangements for words with all letters different, words with two letters the same  and the words with three letters the same and so on.

  • Part 1

I am going to look at the number of different arrangements of the letters of Lucy’s name.

LUCY

As it shown above, there are 24 different arrangements in four letter word.

        C                Y                

        U

                             Y                        C

                              U                Y

L        C

      Y                U        

                               

                               C                U

Y

                               U                C

A tree diagram can also be used as I have used above. Here we have used “L” as a first letter; however the same process can be used with other 3 letter, so we can multiply the number of letter by the number of possibilities, 6*4=24

 Each starting letter has six possibilities. Therefore four letters altogether 6+6+6+6=24

You will get 24 possibilities altogether.

First of all we have got 4 possibilities (4 letters) and then 3 possibilities(by leaving first letter “L”) and then 2 possibilities 9by leaving first two letters “L”U”) after that one possibility which is the last letter . Therefore 4*3*2*1=4! Which is equal to 24, where as we can see this from tree diagram (which is shown by using different colours)

  • Part 2

I am going to investigate the number of different arrangements of the letters of EMMA’S name.

EMMA

There are twelve arrangements for a word with four letters with the two letters the same, and as the two letters are the same; the number of arrangements is halved. This is because

For example: EMMA

                  EMMA                These are two different arrangements but in this case, it indicates one because even though when we coloured and swap the letter M, the letters are the same and therefore the arrangements are the same. So we consider that as one. The same thing is happening in the all twelve arrangements (shown at the top) it is suppose to be 24 but we consider as twelve arrangements, When the 24 arrangements are paired up.

Therefore four letters all together and two letters the same, has 12 arrangements.

Now I am going try the same word EMMA with the tree diagram

        

        M                A                

        M

                             A                        M

                              M                A

E        M

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     A                M        

                               

                               M                M

A

                              M                M

These different colours indicate that each colour paired equal to one. So each two boxes are in one colour and there are three different colour , so three different arrangements.

  • For example: EMMA

                ...

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