INAKR IKRAN IANRK IRKAN
INRAK IKNRA IAKRN IRANK
INRKA IKNAR IAKNR IRAKN
For this, there are 120 possible combinations
RESULTS
Table of Results
Graph of Results
FORMULA
Now I will work out a formula that will let me find out how many combinations there are just from looking at how many letters in a name there are.
For a 1 letter name, you multiply it by itself: 1x1=1
For a 2 letter name, you multiply it by 2 and 1: 2x1=2
For a 3 letter name, you multiply it by 3, 2 and 1: 3x2x1=6
For a 4 letter name, you multiply it by 4, 3, 2 and 1: 4x3x2x1=24
For a 5 letter name, you multiply it by 5, 4, 3, 2 and 1: 5x4x3x2x1=120
This is how you receive your results, you just simply just times it by each number smaller than the number of letters there are in the word you are investigating. This is called “factorising”. The algebraic formula for this method is
C=N!
C= Combinations
N= Number of letters in word
!= Factorial
Using this formula I have found out, I shall now make some predictions.
PREDICTIONS
My predictions are as follows
LUCY
Part of our investigation is to find how many combinations there are in the name Lucy. I will also do this to check whether my previous results were correct
LUCY UCLY CULY YLCU
LUYC UCYL CUYL YLUC
LCYU UYCL CLYU YUCL
LCUY UYLC CLUY YULC
LYCU ULYC CYUL YCUL
LYUC ULCY CYLU YCLU
There are 24 different combinations here, which prove that my previous results were correct. I shall now work out a formula that lets me find out how many combinations there are in names with 2 letters the same, 3 letters the same and 4 letters the same.
NAMES WITH 2 LETTERS THE SAME
I will now work out a formula that will let me work out how many combinations there are in words with 2 letters the same. I will use the following names:
2 letter name
AA
3 letter name
AAL
4 letter name
ALAE
5 letter name
ANNIE
Also, I will look at the name Emma as part of our investigation, and to help prove my results and formula correct.
RESULTS
2 letter name
AA
For this, there is only 1 possible combination.
3 letter name
AAL
LAA
ALA
3 possible combinations are possible with this name
4 letter name
ALAE LAEA EAAL
ALEA LAAE EALA
AAEL LEAA ELAA
AALE
AELA
AEAL
For a 4 letter name, there are 12 different combinations
5 letter name
ANNIE ENNIA INNAE NNAEI NEANI
ANNEI ENNAI INNEA NNAIE NEAIN
ANEIN ENAIN INENA NNEAI NEIAN
ANIEN ENANI INEAN NNEIA NEINA
ANENI ENINA INANE NNIAE NENAI
ANIEN ENIAN INAEN NNIEA NENIA
AENIN EANNI IENNA NIEAN NAINE
AEINN EANIN IENAN NIENE NAIEN
AENNI EAINN IEANN NINAE NAENI
AINEN EINNA IANNE NINEA NAEIN
AINNE EIANN IAENN NIAEN NANIE
AIENN EINAN IANEN NIANE NANEI
For a 5 letter name there are 60 possible combinations.
RESULTS
For names with 2 letters the same, I have done some examples, and am now going to show my results.
Table of Results
Graph of Results
FORMULA
I have now worked out a formula and will now explain it:
For a 2 letter word: 2x1=1
2
For a 3 letter word: 3x2x1= 3
2
For a 4 letter word: 4x3x2x1=12
2
For a 5 letter word: 5x4x3x2x1=60
2
So first of all, you factorise like you did in the 1 letter the same formula, and then you divide it by two. The reason you divide it by 2 is because there are two letters the same. Because there are 2 letters the same, the order that you change the word in changes, meaning you have less combinations.
Predictions
I shall now make a few predictions based on my formula and results.
EMMA
To make sure my results are correct, I will now investigate the word Emma. It is also part of our objective to find out about the word Emma.
EMMA MMEA
EMAM MMAE
EAMM MEAM
AMME MEMA
AMEM MAEM
AEMM MAME
For this, there are 12 combinations, which prove my previous results were correct, as well as the formula.
If I was to colour reach “E” separately, I would get 24 results. This is because the letters that are the same would look the same, but technically wouldn’t be the same. However, I didn’t do this, which meant that I get 12 results. Also, this is why I have to divide the result by 2, otherwise the result would be like there were no letters the same in the name.
Names with 3 letters the same
Now I will look at words that have 3 letters the same, and work out a formula for this like I have done with the previous 2. The names I will use are:
3 letter name
AAA
4 letter name
AAAL
5 letter name
LAAAB
RESULTS
3 letter word
AAA
For this, there is 1 possible combination
4 letter word
AAAL
ALAA
AALA
LAAA
For this word, there are 4 possible combinations.
5 letter word
LAAAB BAAAL ABAAL AAABL
LBAAA BLAAA ABLAA AAALB
LABAA BALAA ABALA AALAB
LAABA BAALA ALAAB AALBA
ALBAA AABLA
ALABA AABAL
For this, there are 20 different combinations
RESULTS
Here are my results
Table of Results
Graph of Results
FORMULA
I have looked at my results, and am now able to work out a formula.
For a 4 letter word: 4x3x2x1=4
3x2x1
For a 5 letter word: 5x4x3x2x1= 20
3x2x1
These results are correct, so therefore my formula is correct. I will now write out this formula in algebraic form.
C=N!/S!
C= Combinations
N= Number of letters
S = Same letters
! = Factorise
Predictions
Based on my formula, I will now make a few predictions
The reason we divide it by 3 factorised, is because there are 3 letters the same, and if we didn’t divide it, it would give the results as if each same letter was different. This means we must divide it by 3 factorised to get the right result.
Names with 4 letters the same
Now I will look at names with 4 letters the same. I will use the following names
4 letter name
AAAA
5 letter name
AAAAL
RESULTS
4 letter word
AAAA
Only 1 possible combination
5 letter word
AAAAL
LAAAA
ALAAA
AALAA
AAALA
5 possible combinations
RESULTS
Table of Results
Graph of Results
FORMULA
Now I will attempt to find a formula based on my results
For a 5 letter word: 5x4x3x2x1=5
4x3x2x1
This is correct, which proves that my results were correct. I shall now show the algebraic formula, which is
C=N!/S!
C= Combinations
N= Number of letters
S = Same letters
! = Factorise
PREDICTIONS
