In this project I had to find out the number of different ways you could arrange any letter words with no repeats, then with one repeat and then find a formula for any number of repeats.

Cuyl                Cyul                Cylu                Culy                Clyu                Cluy

Ulcy                Ucly                Uycl                Uylc                Ucyl                Ulyc

Ycul                Yclu                Yluc                Ylcu                Yucl                Yulc

I thought this answer was 24 because there were 4 different letters and 6 different ways to arrange it with the same first letter and so 6x4 is 24

The next name I did was Chloe. I came up with 120 ways of arranging it. For each different first letter there were 24 different ways of arranging the other letters, as ;

Chloe        Chleo        Choel        Cheol        Cheol        Chelo        Cloeh        Clohe        Cleho        Cleoh        Clhoe        Clheo

Coehl        Coelh        Colhe        Coleh        Cohel        Cohle        Ceolh        Ceohl        Cehlo        Cehol        Celho        Celoh

So there would be a total of 24 x 5 = 120 ways of arranging the letters.

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The Formula

Next I wrote out a table with my results in it:

All Different

Letters                        Ways

1. 1
2. 2
3. 6
4. 24
5. 120

Then with this table I worked out a  formula for words with no repeats. Say I was trying to find out the number of ways of arranging a word with 6 letters in it, instead of writing them all out I'd do 1x2x3x4x5x6. My teacher told me that to do this simply I would do 6! On a scientific calculator.

One Repeat

The next thing I had to do was a word with one repeat in it, the first word I did was Lee I came up with three ways of arranging it

Lee

Ele

Eel

Then I did the word Emma I found 12 ways of arranging it.

Emma                Emam                Eamm                

Mmae                Mmea                Maem                

Meam                Mame                Mema

Amme                Amem                Aemm                

Next I did the word Hello and I found 48 ways of arranging this. I then drew out a table of my results

The formula

One Repeat

Letters            Ways

2. 1
3. 3
4. 12
5. 48

I then worked out that this table compared to my last table was very similar

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All Different

Letters                        Ways

1. 1
2. 2
3. 6
4. 24
5. 120

I saw that all the results when one of the letters was repeated was half what it was when none of them was. For example a word with five letters in it with no repeats had 120 ways of arranging it but with one repeat it had 48 ways.

Formula For All Different Number Of Repeats

I then decided that as for one repeat it was the first answer divided by two for two repeats it must be divided by 6 (divided by two and then that answer divided by three). Then for three repeats it would be divided 24 (divided by two then that answer by three and then that answer by four). And so on.

Conclusion

As this was my first piece of Maths coursework I found it quite tricky, but after I got over that I managed to do quite well at it. I however, did find this piece of coursework interesting and challenging and I'm looking forward to my next piece.