Here are some other 4-letter names where one has 2 letters the same, and the other one all letters are different. I am doing this to test that the results are the same as Emma and Lucy.
Arrangements for Elly:
There are 12 arrangements for Elly’s name.
Arrangements for Mark:
There are 24 different arrangements for Mark’s name.
As I have found that there were 24 arrangements for a 4 letter word with all different letters and that there were 6 ways beginning with one of the letter, I predict that there will be 120 arrangements for JAMES, 24 for j, 24 for a, 24 for m; etc. 120 divided by 5 (number of letters) equals 24. Previously in the 4-letter word, 24 divided by 4 equals 6, the number of possibilities there were for each letter.
My prediction was correct as I have calculated the answer and it matches my prediction.
For Aaron, where there are 2 letters the same in a 5-letter word, the number of different arrangements would be half of 120 which is 60 because you can’t have the same arrangements twice for both As
Arrangements for Aaron:
As a general formula for names with n number of letters all different I have come up with a formula. With Lucy’s name: 1x2x3x4 = 24. With James: 1x2x3x4x5 = 120. However this is expressed as factorial. There is a button on most scientific calculators with have embedded this factorial button feature generally sowing as an exclamation mark. All it does is save the time of having to put in to the calculator 1x2x3x4x5x6x7… etc. You just put in the number and press factorial and it will do 1x2x3… until it gets to the number you put in. If I press (lets say the number of letters all different) factorial 6, it gives me 720, witch makes sense because 720 divided by 6 equals 120 which was the number of arrangements for a 5 letter word and it continues to fall in that pattern.
Table of results:
To find out the number of arrangements of n number letters with 2 letters the same, you simply press the factorial button and get your answer, then divide it by two and you will get the correct answer.
To make it a bit easier instead of using letters as such I will use x’s and y’s (any letter). I will start with xxyy:
This is 4 letters with two the same
There are 6 arrangements
What if I had xxxyy?
x=a, y=b
There are 10 different arrangements for this instance.
What if I had an arrangement of xxxxy?
x=a, y=b
There are 5 different arrangements for this instance.
If I go back to xxxyy; there are 3 x’s and 2 y’s in a total of 5. As each letter has its own number of arrangements i.e. there were 5 beginning with x, and 5 beginning with y, I think that factorial has to be used again. Also in a 5-letter word there are 120 arrangements and 24 arrangements (120 divided by 5) for each letter. I came up with; the number of total letters factorial, divided by the number of x’s, y’s etc factorised and multiplied.
Formula equals n=!
