In this piece of coursework my initial aim is to investigate how many different combinations there are for four letters (e.g. ABCD).

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Jeremy Beales                /

Maths Coursework- Matt’s First Theorem

In this piece of coursework my initial aim is to investigate how many different combinations there are for four letters (e.g. ABCD), I also intend to develop this to investigate the way in which by altering the letters to form other kinds of combinations (e.g. ABCC or AAB) the number is affected. Once I have found the general formulae, I will apply these to harder situations and this is what I am aiming to do. I am trying to find the general formulae which can be applied to all situations we set about this by looking at the simplest scenario first i.e. one letter (e.g.A) moving on to harder problems and by the end I hope to be able to find the possible arrangements for any given word. I will do this by using tables and lists of my results to show the possible combinations and make it easy to compare them and to spot the pattern and try and turn this into a general formula. Once the initial formulae have be en discovered I think that it should be much easier to determine the harder formula, as I will not need to write out as many tables, to work out these formulae  

Results-

Single different letters-

1 letter- A

2 letters- AB

                BA

3 letters- ABC

               ACB

               BAC

               BCA

               CAB

               CBA

No. of letters                 1         2         3            

No. of combinations     1         2         6

This gives the formula =n! - Where n equal the number of letters

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Explanation-

This is because once you have picked one letter there are then only two more letters and then one letter. This means that you get 3x2x1 and this gives you 6 which is equal to n!

This formula will allow me to work out the number of combinations of any word without a repeated letter by using this basic idea I will be able to modify it in order to discover the more complicated combinations, for example words with multiple repeats

One Repeated Letter-

2 letters- AA

3 letters- AAB

        ...

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