GCSE Mathematics: Emma's Dilemma

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GCSE Mathematics: Emma's Dilemma

Emma and Lucy are playing with arrangements of their names. One arrangement of LUCY is L.U.C.Y. Another arrangement is Y.L.C.U. We are investigating the number of different arrangements of Emma and Lucy's names. Then we shall move onto the number of different arrangements of various groups of letters.

Firstly, I shall list all the possible combinations for the word "LUCY":

LUCY LCUY LYUC LYCU LUYC LCYU UCYL UYCL ULCY ULYC

UYLC UCLY CYLU CYUL CULY CUYL CLYU CLUY YCUL YCLU

YUCL YULC YLUC YLCU

= 24 Combinations

I listed all these combinations by using the following method:

Step 1

Arrange the listing process into 4 stages:

1 _ _ _

2 _ _ _

3 _ _ _

4 _ _ _

Step 2

You start off with the original word: 1234

Acquire the combinations of the last two numbers first and you end up with 1243.

Now that you have 1243, do the last three numbers and try the different possibilities:

423 1432 1342 1324.

Because the number 2 has been the first number of last three numbers, we don't do it again.

Step 3

Now that we have listed all the arrangements beginning with 1, we do the list of arrangements with 2 in the beginning:

Start off with 2134 and do same thing to it, it will look like this:

2134 2143 2431 2413 2314 2341

Step 4

Now that we have established the different arrangements of 2, we go ahead and do 3 at the beginning:

3124 3142 3241 3214 3412 3421

Step 5

Now that we have established the different arrangements of 3, we go ahead and do 4 at the beginning:

4123 4132 4231 4213 4312 4321

By using the above method you will definitely acquire all the different arrangements of a four letter word with different letters.

Now I shall list all the possible combinations for the word "EMMA". But there is a problem! There are 2 of the same letter (M) in "EMMA". To start off with, I shall use the same method that I used with the word "LUCY".

EMMA EMAM EMMA WMAM EAMM EAMM MMAE MAME MEMA

MEAM MAEM MMEA MAEM MAME MMEA MMAE MEAM MEMA

AMME AMEM AMME AMEM AEMM AEMM

=24 Combinations

Obviously seeing as "LUCY" & "EMMA" are both 4 letter words; by using the same process I acquired the same amount of results (24). There is however, another method. I could treat both the M's as one collective term. Therefore if you swapped them around, you would get the same word or a repeat of a previous combination, so you cannot include it in your list of combinations. So by treating both the M's as 1 collective term I would NOT find 24 combinations. How many would I find?

I shall list all the possible combinations of "EMMA" using this method:

EMMA EMAM EAMM MMEA MMAE MAEM MAME AMEM AMME AEMM MEMA MEAM

= 12 Combinations

I acquired 12 combinations by using this method. This is because in the previous case when we found 24 combinations, each M was treated as if it was a different letter, therefore allowing us to find more arrangements. Whereas in this method we could not swap the M's around as we could do before. Now that we have investigated 4 letter words, let's move onto 5 letter words:

I am going to number each letter and list the combinations as numbers. The 1st letter of the word will be numbered as 1, 2nd as 2, 3rd letter as 3 etc.

2345 13245

2354 13254

2435 --- 6 arrangements 13452 --- 6 arrangements

2453 13425

2534 13524

2543 13542

4253 15423

4352 --- 6 arrangements 15324 ---- 6 arrangements

4523 15342

4532 15243

4325 15234
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Have you noticed that the combinations of last 4 numbers added up equal 24, so if we multiply 24 by 5, and get 120, then 120 should be the total of arrangements.

Carry on and investigate if a number has 6 figures, then the total of different combinations should be 120 times by 6, and get 720, and 720 should be the total of arrangements.

Carry on, if a number has 7 figures, then the total of different combinations should be 720 times by 7, and get 5040, the total of combinations should be 5040.

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