How many arrangements will there be if I had a word with three letters the same? For example saaa:-
From the table above, we can see that there are 4 different arrangements. Every time the letter S is moved along one place, there is another word made to fit into the total arrangements.
How many arrangements would there be if there is a four letter word with two different letters? For example- xxyy:-
Looking at this table, it is obvious to see that there are 6 arrangements. So far in this coursework, I have solved the different number of arrangements for words with 4, 6, 12, 24 arrangements.
I will need a formula to help me find how many arrangements there are when a word has a lot of letters, repeated or not.
I am now going to find out how many arrangements there are for a five letter word with no repeated letters like SMART.
If I divide 24 by 6 it will give me an answer of 6, this is the number of arrangements a 3 letter word with no repeated letters has. If I multiply 24 by 5, it ought to give me the number of arrangements for SMART. Therefore I predict that there will be 120 different arrangements for SMART.
As there is no space for me to put a table with 120 arrangements in it, I will have to rely on my prediction until I make a formula and only then I can prove this prediction right or wrong. But below is a short table of how I would start of expressing how SMART should be arranged:-
As the table gets larger, I can see that there is going to be 5 horizontal rows and 24 vertical columns. This will give me a grand total of 120 boxes which is the number of arrangements for SMART.
Now I can predict how many arrangements there will be for a 6 letter word which has all different letters. This will be calculated by 120 x 6 = 720 arrangements. Note that I got the 120 from my previous answer of arrangements which was the number of arrangements of a 5 letter word with no repeated letters. If I was to calculate the number of arrangements of a 6 letter word using a simple method, it would be written down as 1 x 2 x 3 x 4 x 5 x 6 = 720 arrangements.
Mathematically, this method is known as a factorial. This is expressed as x!
I will now summarise all of my arrangements up by using the factorial method:-
JON = 3! = 1 X 2 X 3 = 6 arrangements
LUCY = 4! = 1 X 2 X 3 X 4 = 24 arrangements
SMART = 5! = 1 X 2 X 3 X 4 X 5= 120 arrangements
Repeated Letters
2 Letters, 2 repeated-
3 Letters, 2 repeated-
4 Letters, 2 repeated-
My General Formula to find out how many arrangements in a word with no repeated letters:-
From all of the arrangements that I have done, I should now be able to make a general formula which can help me find the number of arrangements in any word with no repeated letters.
Below is a table which will help me to express my thoughts on how to do it:-
To prove my formula and to know what my formula is, I will do the following:
As you can see that there is a pattern here. Each formula increases by a one (N + X) factor. To make my formula work, N must equal 1. This will be explained using an example.
REMEMBER N MUST EQUAL 1 (N=1)
If I want to find out the number of arrangements of a word with 4 words which do not have any repeated letters I will calculate the number of arrangements by:-
(N) (N+1) (N+2) (N+3) (N+4)= 1 x 2 x 3 x 4 = 24 arrangements.
Conclusion
As you can see from my method of making a formula, it did not turn out correctly. This is because my formula turned out to be exactly the same as the factorial formula. As there is no quadratic formula because it is a factorial, the only formula to work out how many arrangements there are in words with no repeated letters is N! This is because (N) (N+1) (N+2) is the same as 1 x 2 x 3 = 6.
Therefore the only formula to work out how many arrangements there are in words with no repeated letters is N!
My General Formula to find out how many arrangements in a word with repeated letters:-
If there is a word with repeated letters, what would the formula be then? I will use a combination of x and y to help me to work out a general formula. I have already used the combination XXYY to express how they should be arranged. But now I will use the XX and YY to design a working formula.
How many arrangements are there for XYXYY?
Here are the arrangements:-
From the table we can see that there are 10 different arrangements. How many arrangements would there be if there is the arrangement YXXXX?
Here are the arrangements:-
From the table, we can see that there are a total of 5 arrangements. This means that the number of arrangements have been halved from the previous XYXYY combination. For a 5 letter word with no repeated letters there are 120 different arrangements.
I believe that I have come up with a general formula which only can be used on words with repeated letters. I feel that if I divide the factorial of the total number of words by the number of repeated words, I will get the number of arrangements.
For example, if I had the word YXXXX and I want to find out how many arrangements there are, I would do the following calculations:-
5!
= 5 different arrangements
4!
We can check on the table above and there are 5 different arrangements. This proves that my formula works. Therefore my formula to find out how many arrangements there are in words with repeated letters is:
N!
A! X B! X C!
Whereas A, B and C= the different number of repeated numbers.
If I want to find out how many arrangements there are for XYYXXXYY by using my formula, it would be calculated by:
8! =
4! X 4!
40320 = 70 arrangements
576
Therefore the arrangements for XXXXXYYXYYXY would be:-
12! =
7! X 5!
479001600 = 792 arrangements
604800
Conclusion
As you can see from my method of making a formula, it did not turn out correctly. This is because my formula turned out to be exactly the same as the factorial formula. As there is no quadratic formula because it is a factorial, the only formula to work out how many arrangements there are in words with no repeated letters is N! This is because (N) (N+1) (N+2) is the same as 1 x 2 x 3 = 6.
Therefore the only formula to work out how many arrangements there are in words with no repeated letters is N!
My other formula which will work out how many arrangements there are in words with repeated letters is:-
N!
A! X B! X C!
To prove this formula, I am going to calculate how many arrangements the word Allan has:-
5! = 30 arrangements
2! X 2!
As this answer is correct, I declare that my formula works
Therefore my two formulas are N! and
N!
A! X B! X C!
