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# Emmas Dilema

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Introduction

## Emma’s Dilemma

In this investigation, we had to find out the number of different combinations that could be made when different amounts and combinations of letters were formed. This includes repeated patterns of letters. Emma and Lucy wanted to find the amount of different combinations each of their names could make.

Emma can be represented by the letters AABC as there is the same amount of letters in both and there is one repeat in both as well. Lucy can be represented by ABCD as it contains 4 different letters with no repeats.

I also had to attempt to find a formula to work out the amount of combinations for other amounts of letters.

To find out the amount of combinations which can be made by non-repeated or repeated patterns of letters,

Middle

2 (AB)

2

3 (ABC)

6

4 (ABCD)

24

Combinations with 1 repeat:

AA: 1. AA

AAB: 1. AAB

2. ABA

3. BAA

AABC:  1. AABC

2. AACB

3. ABAC

4. ABCA

5. ACBA

6. ACAB

7. BAAC

8. BACA

9. BCAA

10. CAAB

11. CABA

12. CBAA

Here is a table of my results:

 Amount of letters in combination Amount of different combinations made by letters 2 (AA) 1 3 (AAB 3 4 (AABC) 12

I can predict the amount of different combinations, which can be made by 5 different letters (ABCDE) and the amount of different combinations, which can be made by 5 letters with 1 repeated pattern (AABCD) by using my equation:

the previous amount of letters

x

Amount of letters in the current combination

My formula is X x Y= N.

This is when X is the amount of combinations made from the previous amount of letters, Y is amount of letters in current combinations and N is the amount of different combinations the current group of letters can make.

Conclusion

the previous amount of letters

x

Amount of letters in the current combination

=

24

x      = 120

5

According to my formula, the amount of different combinations produced by 5 different letters is 120.

Now I need to check if the formula works for repeated patterns of letters.

 Amount of letters in combination Predicted Amount Amount of different combinations made by letters 2 (AA) N/A 1 3 (AAB) 3 3 4 (AABC) 12 12

By looking at the information contained in the table, we can see that my formula works for the results I have collected. If it is correct for future amounts of letters, I can predict the amount of different combinations 5 letters with a repeat (AABCD) will make.

the previous amount of letters

x

Amount of letters in the current combination

=

12

x        = 60

5

According to my formula, the amount of different combinations produced by 5 letters with a repeat is 60.

This student written piece of work is one of many that can be found in our GCSE Emma's Dilemma section.

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## Here's what a star student thought of this essay

3 star(s)

### Response to the question

Response to question - To what extent has the student answered the set question? How explicit is their response?
Overall a good piece of work, with good mathematical vocabulary where required. You have explained their methods well by showing all ...

### Response to the question

Response to question - To what extent has the student answered the set question? How explicit is their response?
Overall a good piece of work, with good mathematical vocabulary where required. You have explained their methods well by showing all the steps from the start to create her proposed formula, however, the piece does not explicitly answer the set question: how many combinations can Emma and Lucy make with the letters of their names? You have explained what they have done to answer the question, but a conclusion to tie everything together and to literally answer the given question would help the marker see your understanding of the task and the methods you have used.

### Level of analysis

Level of analysis - To what extent does the writer show appropriate analytical skills for this level of qualification? Have they made evaluative judgements using suitable evidence? Have these examples been developed throughout the response and has an appropriate conclusion been reached?
You have used sufficient mathematical vocabulary, but more could improve the piece if used correctly. You have used evidence to support their reasoning and proposed formula and tested it to see if it works as expected to a suitable degree. This has been explained well and proven to help answer the question, but has not been fully brought to a conclusion. Just a sentence or two to say "Emma can make 24 combinations from the letters in her name, which I have calculated from my AABC listsÃ¢â‚¬Â¦Ã¢â‚¬Â and so on to round the piece off nicely, just to spell out the fact youÃ¢â‚¬â„¢ve answered the set question.

### Quality of writing

Quality of writing - Is the writing accurate in terms of spelling, grammar and punctuation? Has the writer used technical terms expected at this level of qualification? To what extent does the writer follow conventions and expectations for written work at this level?
The spelling and grammar is of a good standard, which is to be expected at GCSE. More mathematical terms would improve the piece but otherwise the writing is of a high standard. The piece could be better laid out, but this may be just a problem of uploading the piece. Everything seems all over the place with extra line breaks and the long list of combinations. A table would condense it and make it easier to read and pick out the required information.
Overall, 3 stars, but would get 4 with a conclusion.

Reviewed by pratstercs 11/02/2012

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