Emma’s Dilemma.

LUCEY

        1) LUCEY        7) LEUCY          13) LYUCE        19) LCYUE

        2) LUCYE        8) LEUYC        14) LYUEC        20) LCYEU

        3) LUYCE        9) LEYCU        15) LYCUE        21) LCEYU

        4) LUYEC        10) LEYUC        16) LYCEU        22) LCEUY

        5) LUEYC        11) LECUY        17) LYEUC        23) LCUYE

        6) LUECY        12) LECYU        18) LYECU         24) LCUEY

Because LUCEY has 5 letters, you can multiply 24 by 5 to get the number of arrangements, this is a much quicker and easier way of counting the different number of arrangements but only works when all the letters are different.

24 multiplied by 5 is 120 therefore LUCEY has 120 different arrangements. I can now correctly predict the number of arrangements a name has no matter how many letters the name has as long as all the letters are different by multiplying the number of letters in a name by the previous consecutive numbers.

To do this, you can use the method of factorial notation which multiplies a number by all the previous consecutive numbers. On a calculator the factorial button looks like this:

                                           !

If you press a number and then !, then it multiplies that number by all the previous consecutive numbers.

Therefore, the formula when all the letters are different is

                                                             A = x!

Where X is the number of letters and A is the number of arrangements.

Now I am going to find out the relationship between the number of letters and the number of arrangements when there are two letters the same.

JO has one arrangement:

        1) JJ

ANN has 3 different arrangements:

1. JJO
2. JOJ
3. OJJ

EMMA, as we know has 12 different arrangements.

Here is a table showing the number of letters and the number of arrangements where 2 letters are the same.

        

Even though there are 2 letters the same, the relationship between the number of letters and the number of arrangements is very similar to when there are no letters the same. 

When there are 4 letters in a word 4x3x2x1=24, you then divide the 24 by

2 to get 12, which is the number of arrangements.

To check this I will use the name JENNY; this has 5 letters with 2 letters being the same.

I predict that JENNY will have 60 different arrangements because

    

                                5x4x3x2x1  = 60

                                        2

1) JENNY        7) JNNEY

2) JENYN        8) JNNYE

3) JEYNN        9) JNYNE                         Because JENNY has 5 letters, by using

4) JYENN        10) JNYEN                       the quicker method of counting, you can

5) JYNEN        11) JNEYN                       say 12 x 5 = 60.

6) JYNNE        12) JNENY

Therefore, my prediction was right. I can now predict the number of arrangements a series of letters has when 2 letters are the same no matter how many letters the name/series has by multiplying the number of letters in the name/series by the previous consecutive numbers and then dividing the total by 2.

Therefore the formula when 2 letters are the same is

                                         

                                                         A = x!

                                                                         2                          Where x is the number of letters and A is the number of arrangements.

This may be because there are 2 letters the same so you divide by 2.

However you may have to divide by 2 because there are 2 letters the same and

2! = 2 so the formula could also be

A= x!

      s!

Where A is the number of arrangements, x is the number of letters and s is the number of letters the same.

To test this out I will use XXXY where X is any letter and Y is any other letter.

If the first theory is right then the number of arrangements should be

4 x 3 x 2 x 1  = 8.

         3

However if the second theory is right then the number of arrangements should be  4 x 3 x 2 x 1  = 4.

          3x2x1    

So here are the arrangements for XXXY:

1. XXXY
2. XXYX
3. XYXX
4. YXXX

As you can see, there are 4 arrangements for XXXY, this shows that my second theory is correct.

 

Now I will investigate what happens when there are 2 different letters which are used more than once for example XXXYY.

Here are the arrangements for XXXYY:

1. XXXYY        6) YXXXY
2. XXYXY        7) YXXYX
3. XXYYX        8) YXYXX
4. XYXYX        9) YYXXX
5. XYXXY        10) XYYXX

As you can see there are 10 arrangements.

Here are the arrangements for XXXYYY:

1. XXXYYY        6) XYXYXY
2. XXYXYY        7) XYXYYX     These are all the arrangements which start
3. XXYYXY        8) XYYXYX     with X. As there are an equal number of X’s
4. XXYYYX        9) XYYXXY      and Y’s. You can multiply the number of
5. XYXXYY        10) XYYYXX    arrangements starting with X by 2 to get 20

Here is a table showing the number of letters, the number of X’s, Y’s and the number of arrangements.

Here, there is definitely a relationship. If you use where there and 4 letter with 3 X’s and 1 Y you do      4 x 3 x 2 x 1          which makes  24  which equals 4 which

                                  (3 x 2 x 1)x(1 x 1)                               4

agrees with the table.

Therefore with XXXXYYYY, there should be 70 different arrangements because

   

      8!       = 70

 4! X 4!

Here are the arrangements for XXXXYYYY:

1. XXXXYYYY        8) XXYYXXYY        15) XYXYXXYY        22) XYYXXYYX
2. XXXYXYYY        9) XXYYXYXY        16) XYXYXYXY        23) XYYYYXXX
3. XXXYYXYY        10) XXYYXYYX        17) XYXYXYYX        24) XYYYXXXY
4. XXXYYYXY        11) XXYYYXXY        18) XYXYYXXY        25) XYYYXXYX
5. XXXYYYYX        12) XXYYYXYX        19) XYXYYXYX        26) XYYYXYXX
6. XXYXXYYY        13) XXYYYYXX        20) XYXXYYXY        27) XYYXYXXY
7. XXYXYXYY        14) XYXXXYYY        21) XYXXYYYX        28) XYYYYXXX

29) XYYXYXYX                  

30) XYXYYYXX              These are all the arrangements which start with X.

31) XYXXXXYY               As there are an equal number of X’s and Y’s, you

32) XYYYXXYX               multiply the number of arrangements so far by 2

33) XYYXXYXY               to get the total number of arrangements.

34) XYXYYYXX

35) XYYXXXYY

Therefore the overall formula is

                                            A =   X!  

                            S!L! 

Where A is the number of arrangements, x is the number of letters, s and l are the number of letters the same.