Emma's Dilemma

HILP                        LIHP

HIPL                        LIPH

HPIL                        LHPI

HPLI                        LHIP

HLIP                        LPIH

HLPI                        LPHI

I have found out that they are 24 different combinations. 4 letters all of them different. I have worked that this is double to EMMA. I have also worked out that with every 2 letters beginning they are 12 combinations so with every one letter beginning they are 6 different combinations.

I will now try 5 letters and find out how many different combinations they are. I predict that they will be 120 different combinations. I predict this because they is 24 combinations for PHIL so if you added another letter you could put the new letter at the beginning of each combination. I will try and find out how many combinations they are with PHILM.

MPHIL                        MILPH                        

MPHLI                        MILHP

MPIHL                        MIHPL

MPILH                        MIHLP

MPLHI                        MIPLH

MPLIH                        MIPHL

MHILP                        MLIHP

MHIPL                        MLIPH

MHPIL                        MLHPI

MHPLI                        MLHIP

MHLIP                        MLPIH

MHLPI                        MLPHI

My prediction was correct they are 24 combinations for every letter beginning they are 24 combinations so 24 times 5 (the number of letters) equals 120.

I have worked out a formula, with PHIL’s name I have found that the total number of combinations was 24 with 4 letters all different 1x2x3x4=24, and with MPHIL they are 5 letters all different, 1x2x3x4x5=120. I have also found an even quicker way to do this on a calculator, they is a factual button that looks like n! Or x!. All this simply does is saving you put 1x2x3x4x5 etc in you just put the number of letters and press the button. The button is called factorial this is found on most scientific calculators. I will now test my formula I will find out how many combinations they are for a 6 letters all different. I will put in my calculator 6 factorial I get the answer 720. This makes sense because 720 divided by 6 equals 120 which was the number of combinations for a 5-letter word and it continues to fall into that pattern.

On the following page is a table showing how many combinations with x number of letters.

 

As you can see in the table the number of combinations dramatically increases.

Now I need to find out a formula for words with 2 letters the same, such as EMMA this has 4 letters and 2 are the same we already know that 4 letters equals 24 so how do we find out how many combinations they are for with repeated letters in a word.  

I will now try a 4 letter word with 2 different

MMAA

MAMA

MAAM

AAMM

AMAM

AMMA

I have found that they are 6 combinations I will now try a 5 letters with 2 different

MMMAA

MMAMA

MMAAM

MAMAM

MAMMA

MAAMM

AAMMM

AMMMA

AMAMM

AMMAM

I have found out that they are 10 different combinations.

I have worked out a logical formula, the number of letters factorial, divided by the number of A’s, M’s factorised. I will now test my formula;

For example I will work out MMMAA they are 5 letters 2 different I would work out, 5 factorised (the number of letters) equals 120 divided by 3 factorised (the number of M’s) multiplied by 2 factorised (the number of A’s) equals 12 so 120 divided by 12 equals 10.

A 4 letter word such as EMMA, this has 3 different letters

1x2x3x4 divided by 1x2 equals 24 divided by 2 equals 12

Below is a table showing the combinations for words with repeated letters;

                                                 

I have tested and proven my formulas and they both work.