I have been given a problem entitled 'Emma's Dilemma' and I was given the following information: 'Emma and Lucy are playing with arrangements of their names

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Emma’s Dilemma

I have been given a problem entitled ‘Emma’s Dilemma’ and I was given the following information:

‘Emma and Lucy are playing with arrangements of their names. One arrangement of Lucy is:

L        U        C        Y

A different arrangement is:

Y        L        C        U

Part 1:

Investigate the number of different arrangements of the letters of Lucy’s name.

Part 2:

Investigate the number of different arrangements of the letters of Emma’s name.

Part 3:

Investigate the number of different arrangements of various groups of letters.’

So basically I had to investigate how the number of permutations for words alters depending on the letters that make up that word, starting off with looking at Lucy and then comparing it to Emma, and after that extending it to look at any word with various letters. I aim to produce a formula to produce the number of permutations.

Part 1- Investigating Lucy:

The name Lucy comprises of four different letters that we can quite simply rearrange to produce a number of arrangements. Firstly I worked out the number of arrangements that I could get from the letters of ‘Lucy’:

LUCY                LUYC                LYUC                LYCU                LCUY                LCYU                ULCY                ULYC                UCLY                UCYL                UYLC                UYCL

CLUY                CLYU                CYLU                CYUL                CUYL                YLUC                YLCU                YUCL                YULC                YCUL                YCLU

I found that the name Lucy can be arranged 24 different times as shown above. This would work for any four letter word as long as all the letters were different. So after realising this, I decided to investigate this further. I then looked at words with different numbers of letters, with all the letters making up the word being different. I came up with the following:

One Letter:

For a word with one letter, there is only 1 combination as shown below:

A

Two Different Letters:

For a word with two different letters, there are 2 arrangements in which the letters can be rearranged, as shown below:

AB                BA

Three Different Letters:

For a word with three different letters, there are 6 arrangements in which the letters can be rearranged, as shown below:

ABC                ACB                BAC                BCA                CAB                CBA

Four Different Letters:

As we were shown from the word Lucy, for a word with four different letters, there are 24 different arrangements in which the letters can be arranged, as shown below:

ABCD                ABDC                ACBD                ACDB                ADCB                ADBC                BACD                BADC                BCDA                BCAD                BDAC                BDCA        

CABD                CADB                CBAD                CBDA                CDAB                CDBA                DABD                DACB                DBAC                DBCA                DCAB                DCBA

Five Different Letters:

For a word with five different letters, there are 120 arrangements in which the letters can be rearranged.

ABCDE        ABCED        ABECD        ABEDC        ABDEC        ABDCE        ACBED        ACBDE        ACDBE        ACDEB        ACEBD        ACEDB

ADCEB        ADCBE        ADBCE        ADBEC        ADEBC        ADECB        AECBD        AECDB        AEBCD        AEBDC        AEDBC        AEDCB

There are 24 arrangements above, all beginning with the letter A. If this process was repeated, with the other letters (B, C, D and E) at the beginning, moving A into the letters that are swapping round, there would be four more lots of 24 arrangements; therefore there would be 120 arrangements in total.  

From the investigation above, I looked for a pattern relating the number of different letters in the word and the total number of different arrangements. I created a table to make it easier for me to view the data:

Table 1


From looking at this table, it looked like there was a pattern linking the numbers. I worked out that the total number of arrangements was worked out by multiplying the number of letters by all the numbers under that number. So if I was to work out the number of arrangements for a 4 letter word, I would multiply 4 by 3 by 2 by 1, as they are the numbers below that number. For a five letter word, you would do the same, so far –        Five Letter Word = 1 x 2 x 3 x 4 x 5 arrangements = 120 arrangements.

This can be explained as there when we have a 5 letter word with different letters, the arrangements are the same as for a four letter word, but with an added letter at the beginning. All the letters can be placed at the beginning and we then get a the number of arrangements for a four letter word with each letter at the beginning. So as there are 24 permutations for a four letter word and there are five choices for the first letter whilst moving the other letters around, we get five lots of 24 arrangements.  

Join now!

This is known as increasing at a factorial rate. Factorials can be written in short with the factorial number followed by an exclamation mark ( ! ). For example four factorial would be expressed as ‘ 4 ! ’.

The factorial of the total number of different arrangements (y) for any different number of different letters (x) is equal to the number of letters, may be written as:

                                Y        =        X!

Part 2 – Investigating Emma:

Part 2 of Emma’s Dilemma is to investigate the number of arrangements for the name Emma. The difference with this name is that ...

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