PART 3: So now I want to see how many ways various group of letters can be arranged. First I will start with 1 lettered word then 2 then 3,4 and then 5 lettered word to see how many ways they can be arranged.

- one lettered word:

a (1x1=1)

It can be arranged only 1 different way.

- Two lettered word:

So (1x2=2)

os

It can be arranged two different ways.

- Three lettered word

Joe oje ejo (1x2x3=6)

Jeo oej eoj

This can be arranged 6 different ways.

- Four lettered word.

See part 1 for 4-lettered word.

It can be arranged 24 different ways. (1x2x3x4=24)

5. Five lettered word.

Kamen akmen mkaen ekamn nkame

kamne akmne mkane ekanm nkaem

kamen akemn mkean ekman nkmae

kamne akenm mkena ekmna nkmea

kaemn aknme mknae eknam nkeam

kaenm aknem mknea eknma nkema

kmnae amken maken eakmn nakem

kmnea amkne makne eaknm nakme

kmaan amekn maekn eamkn namke

kmane amenk maenk eamnk namek

kmean amnke manke eankm naekm

kmene amnek manek eanmk namke

kenam aekmn mekan emkan nmkae

nenme aeknm mekna emkna nmkea

keamn aemkn meakn emakn nmake

keanm aemnk meank emank nmaek

keman aenkm menka emnla nmeka

kemna aenmk menak emnak nmeak

kneam ankme mnkae enkam nekam

knema ankem mnkea enkma nekma

kname anmke mnake enakm neakm

knaem anmek mnaek enamk neamk

knmae anekm mneka enmka nemka

knmea anemk mneak enmak nemak

It can be arranged 120 different ways. (1x2x3x4x5=120)

This shows that the more letters a word has the more different ways it can be arranged.

Example 1 lettered word can be arranged 1 way, 2 lettered word can be arranged 2 different ways, and so on.

The theory that I found in part one would work with every word only if it doesn’t have any identical letters.

Here are the results:

From this I found a formula to find how many ways a word could be arranged.

The formula I have found is A = n! (when there is no repeated letters), A is for arrangements, n! stands for the factorial, which is n x (n-1) x (n-2) x (n-3)…until 1, so in 5 lettered word the arrangements will be 5 x 4 x 3 x 2 x 1 = 120, it is also the same thing if it is done backwards example of a five letter word, 1 x 2 x 3 x 4 x 5 = 120. For example if I want to calculate how many ways a four lettered word can be arranged, this is how it will be done, 1 x 2 x 3 x 4 =24.

As we can see from above results are right, so now we are sure that this formula will work for any number lettered word.

I can see another formula that might work.

From this I found out a formula, the formula is N X result for N- 1, here "n" stands for the number of letters in a word , multiplied by the number of arrangements that can be made with a word that has a letter less. for example if I want to find the arrangements of a 5 lettered word, I will have to multiply 120 by 24, the reason why I will have multiply by 24 is because a four lettered word (no double leters) can be arranged only 24 different ways.

The formula is N x result for N-1. To see if it works, lets try with a 3 lettered word.

N x result for N-1

3 x 3-1 =2(because a 2 lettered word can be arranged twice)

3 x 2 = 6

Yes it works lets try with a 5-lettered word

N x result for N-1

5 x 5-1= 24 (because a 5 lettered word can be arranged 24 different ways).

This proves this formula works.

But what about a 1 lettered word, the formula would not work. Example it is obvious a 1 lettered word can be arranged only one way, but the formula will give 0 as the answer.

N x (n – 1)

1 x ( 1 – 1)

1 x 0 = 0

One thing with the above formula is that if a word contains a double lettered word, the above formula cannot calculate the number of arrangements a word will have if it has more than two same letters.

Example the word abb, abb

bab

bba

It can be arranged three times, the above formula will be N x result for (n-1)

3 x (3-1=2)

3x2=6

From this we can see that the above formula cannot calculate the number of arrangements for a word that has two or more repeated letters. So this means that the above formula will only work with a word that has no repeated letters.

Before finding a formula to calculate the number of arrangement for a word that contains double letters, I will have to find out the results for words containing double letters.

- 2 letters of the same in a three-lettered word.

Ana

Aan

Aan

It can be arranged 3 times.

4 lettered word containing 2 letters of the same.

Emma mmea mmae amme

Emam maem amme amem

Eamm meam mame aemm

As I mentioned before if a word has 2 identical letters it can be arranged less differently if the letters are different, example Lucy can be arranged 24 different ways and Emma can be arranged 12 different ways.

From this I can see a formula , the formula is, A = n!/ s!, the beginning part of the formula has been explained before, the last part means the same thing also but instead of multiplying you divide because of ‘/’. This means you will have to find the number of letters a word contains multiply from 1, up to the numbers it contains, then divide by 2, lets try with a 3 lettered word, 1 x 2 x 3 = 6 = 3.

1x2= 2 2

Now lets try with a 4 lettered word, 1 x 2 x 3 x 4 = 24 = 12

1x2= 2 2

But what if a letter has more then 2 same letters this formula doesn’t work, example of a 4 lettered word,

1 x 2 x 3 x 4 = 24 = 12

2 2

Aaab

Aaba

Abaa

Baaa

It can be arranged 4 different ways.

So now this leaves me to find another formula that will calculate how many ways a word can be arranged wither if it contains two or more identical letters.

From this I have found a formula, the formula is A = n! / z! y! ( when there are 2 letters repeated in a word 2 or more times ), this means multiply from 1 up to the number of letters then divide by multiplying from 1 again up to the number of same letters.

To see if this formula works lets try with a four lettered word containing three letters the same,

1 x 2 x 3 x 4 =24 = 4

1 x 2 x 3 = 6

abbb

bbba

bbab

babb

Now lets try with a 3 lettered word with 2 the same, 1 x 2 x 3 = 6 = 3

1 x 2 2

abb

bba

baa

Yes, this formula works with any word that contains any amount of same letters, as you can see from above.

There is one more thing that has left me thinking is that if a word has one letter repeated twice and another letter is also repeated twice. The above formula will not work. Example the word aabb.

aabb 1 x 2 x 3 x 4 = 24 = 0

1 x 2 x 3 x 4 = 24 =

aabb

abab

abba

bbaa

baab

The above formula calculates it can be arranged 0 times, so now I have to find another formula that will enable us to calculate how many ways a word an be arranged with the above problem.

The formula that I have found is A = n! / z! y! ( when there is 2 letters repeated in a word 2 or more times ), this means to multiply from starting from 1 up to the numbers it contains then separate the different identical letters and multiply each of them by the number of times it has occurred. Example with the word aabb.

As above we know the word aabb can be arranged 6 different ways, so now lets use the formula to see if the formula works. The formula for the above word would be

1 x 2 x 3 x 4 = 24 = 6

(1x2) x (1x2) = 4

The above formula does work as we can see from above. Now lets try with another word, which has five letters with containing four letters, the same with two pairs.

abbcc bbcca ccbba

abcbc bbcac ccbab

acbcb bbacc ccabb

accbb babcc cacbb

acbbc bacbc cabcb

abccb baccb cabbc

bccba cbbca

bccab cbbac

bcbac cbabc

bcbca cbcba

bcabc cbacb

bcacb cbcab

The formula would be for this, 1 x 2 x 3 x 4 = 120 = 30

1x (1x2)x(1x20)= 4

Yes, the formula does work.

If there is a word that has two letters the same and occurs more than two times, the formula is A = n!/ z! y! x!, and so on.

Now I cannot think of any problem that would arise in finding how many ways any words can be arranged, also any word that wouldn’t fit in to any of the above formulas.

From this investigating I learnt many things, for example the more letters a word contains the more different ways it can be arranged, I learnt how to find formulas etc. I really enjoyed doing this investigation , and the main benefit I got from this investigation is that I will never have any problem in calculating how many different ways a word can be arranged.