Sam
Sam
Sma
Ams
Asm
Mas
Msa
Sam is a 3 letter word, and all the letters are different, giving us a total of 6 combinations.
JJ
JJ
JJ is a letter name and both of the letters are the same; therefore there is only 1 combination of this name.
Jo
Jo
Oj
Jo is a 2 letter name and both of the letters are the same giving us a total of 2 combinations.
The table above shows us, to get the total number of combinations when 2 letters are the same, is, divide the number of letters by 2 factorial. To get the total number of combinations, when 3 letters are the same, is, divide the number of letters by 3 factorial. To get the total number of combinations when 4 letters are the same, is, divide the number of letters be 1 factorial. And to get the total number of combinations when 5 letters are the same, is, divide the number of letters by 5 factorial.
To work out the total number of combinations, when all the letters are different, we will have to use factorial. Factorial is a number multiplied by the previous consecutive numbers. This is shown using an exclamation mark. So the formula for this is:
n!
n is the number of letters !
For example:
9 letters - 9! = 362880
8 letters - 8! = 40320
7 letters - 7! = 5040
6 letters - 6! = 720
5 letters - 5! = 120
4 letters - 4! = 24
3 letters - 3! = 6
2 letters - 2! = 2
All these have been proved in previous arrangements, which shows that my formula does work.
xxyy
xxyy
xyxy
xyyx
yxxy
yxyx
yyxx
xxyy is a 4 letter name, with6 different arrangements.
xxxyy
xxxyy
xxyxy
xxyyx
xyxyx
xyxxy
xyyxx
yyxxx
yxxxy
yxyxx
yxxyx
xxxyy is a 5 letter name, and there are 10 different arrangements for this name.
xxxxy
xxxxy
xxxyx
xxyxx
xyxxx
yxxxx
xxxxy is a 5 letter word ands there are 5 different arrangements name.
If I go back to xxyy; there are 2 x's and 2 y's in a total of 4 unknowns. As each letter has its own number of arrangements, I think that factorial has to be used again. Also in a 4-letter word there are 24. I divided the issue involved I had a go at trying to work out a formula, and I came up with:
n!/x!y!
n is the number of letters !
x is the number of x’s !
y is the number of y’s !
For example:
5 letters – xxxxy = 5! / 4! x 1! = 120 / 24 = 5
4 letters – xxyy = 4! / 2! x 2! = 24 / 4 = 6
all different – vwxyz = 4! / 1!x1!x1!x1!x1!x1! = 24 / 1 = 24
All of these formula have been proved in previous examples, which shows that my formula does work.
