I am going to begin by investigating the number of arrangements in Emma’s name.
Emma
EMMA
EMAM
EAMM
AMME
AMEM
AEMM
MEAM
MAEM
MMEA
MMAE
MEMA
MAME
I have found 12 different combinations for Emma’s name. I will now investigate the number of arrangements of Lucy’s name.
Lucy
LUCY
LUYC
LCUY
LCYU
LYCU
LYUC
ULYC
UYLC
UCYL
UCLY
ULCY
ULYC
CYLU
CYUL
CULY
CUYL
CLYU
CLUY
YLCU
YLUC
YCUL
YCLU
YULC
YUCL
I found out that there are 24 different arrangements of Lucy’s name. This is double the amount of Emma’s name.
There are 4 letters in both Emma’s and Lucy’s name but Emma has two letters that are the same. I will further investigate how having same letters affect the number of combinations.
First, I will explore names with different letters. I will use the names Jo, Sam, Lucy and Emily.
*Previous results show that Lucy has 24 arrangements.
From my results, I have found a pattern. In Lucy’s name, there are 4 letters and 24 possible arrangements. 24 is equal to 4x3x2x1. Therefore, in order to work out the number of arrangements for a name with different letters, you must multiply the number of letters by every whole number between itself and 1 inclusive. If the number of letters was n, this would be the formula:
n(n-1)(n-2)(n-3)…x3x2x1
This form of multiplication is called factorial. The symbol for factorial is ! Therefore, the formula is:
n!
This means that the number of arrangements for the name Emily is 5! which equals 120.
Next, I will investigate names where two letters are the same.
Here are my results:
Compared to my previous results, these results are exactly half. This means that in order to find the number of arrangements for names with 2 same letters, you must divide our original formula, n! by 2. Therefore the formula for names where 2 letters are the same is:
n!
2
2 is equal to 2x1 so, the formula is:
n!
2!
This means that the number of combinations for the name, MMABC is 5! divided by 2! which equals 60.
I will now investigate names where 3 letters are the same.
Here are my results:
In comparison with my original results, the numbers of arrangements are divided by 6. 6 is also equal to 3! Which is 3x2x1. So, the formula for names where 3 letters are the same is:
n!
3!
This means that the number of combinations for the name MMMAB is 5! divided by 3! which equals 20.
I will now investigate names where 4 letters are the same.
Here are my results:
When I compare this with my original results, the numbers of arrangements are divided by 24. 24 is also equal to 4! Which is 4x3x2x1. Therefore, the formula for names where 4 letters are the same is:
n!
4!
This means that the number of combinations for the name MMMMAB is 6! divided by 4! Which equals 30.
I have found out that in order to find out the number of permutations for names with same letters, you must divide the number of letters factorial by the number of same letters factorial. For example, the name Igein, this name has 5 letters and 2 same letters, so the answer would be:
5!
2!
This equals 60.
The final part of my investigation will involve sets of same letters e.g. Anna.
There are 6 combinations for the name MMLL and 10 for MMMLL. I have discovered that if we divide the number of letters factorial by the number of the 1st set of same letters multiplied by the number of the 2nd set of same letters, it will give us the number of arrangements. If the number of Ms = y and the number of Ls = x, this would be the formula for working out the number of combinations for names with sets of same letters.
n!
y!x!
This formula will work for any amount of sets of same letters. For example, the name Hannah which has 6 letters and 3 sets of 2 same letters. The answer would be:
6!
2!2!2!
The answer is 90.
Conclusion: During this investigation, I found a general formula to work out the permutations of names.
Names with different letters when n equals number of letters:
n!
Names with same letters when n equals number of letters and s equals the number of same letters:
n!
s!
Names with sets of same letters when n equals number of letters:
e.g. ANNA
N= a and A=b
n!
a! b!
These formulas that I have found will enable me to work out the number of permutations for any name.
Written by Emily Kho
