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Emma’s Dilemma

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Emma’s Dilemma

In this investigation I will try and work out how many combinations I can make out of any type of word.

4 letters

The name EMMA has 4 letters, with 2 letters the same.  I will experiment how many combinations I can make with this name.

I find it easier to try and make each combination using one letter, then another etc.

EMMA        MEMA     MEAM    AMME

EMAM        MMEA     MAEM    AMEM

EAMM        MMAE     AEMM

The name LUCY has 4 letters, which are all different.





The name SASS has 4 letters, 3 of which are the same.



The name AAAA has 4 letters, which are all the same.

AAAA has no variations.

3 letters

The name LOU has three letters, all different.


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image04.png I shall multiply 5 X 24 to get the combinations for 5 letters. I will then half the answer I get from this to find out the combinations for 5 letters with 2 the same. I’ll divide this answer to find the combination for a 5 letter word with 3 the same, and so on.

No. of letters

All different

2 the same

3 the same

4 the same

5 the same














5X24= 120

120image03.png 2=60

60 image03.png 3=20

20 image03.png 4=5


From these results I have now found out how this works. The number of combinations is narrowed down each time there is more that 1 letter the same, or 1 letter is chosen to begin.

This is what I mean:

E.g. 2 letter word

2 choices for letter, multiplied by one choice for letter = 2

3 letter word


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E.g. 10 ! = 10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1

        10!  = 3, 628, 800

I can therefore say that a 10 letter word with all of the letters different could make 1, 628, 800 combinations of itself.

I have come up for this equation for finding the combination of an n letter word, with all letters different.

  N! = n x (n - 1) x (n - 2) x (n - 3) x (n - 4) x (n - 5) x (n - 6) x (n - 7) … etc.

If 2 letters were the same, the equation would be:

n! image03.png2

And if 3 letters were the same, it would be:

n! image03.png 2 image03.png 3                                                                                                         …etc.

This concludes my coursework, as I have now found how many combinations you can get from an n letter word, even when the word has 2 letters the same, 3 letters the same, 4, or 5 …etc.

- Louise Manly

...read more.

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