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  • Level: GCSE
  • Subject: Maths
  • Word count: 1682

permutations & combinations

Extracts from this document...


Sushant Goel                                           Maths Coursework                               Form: 10D

                                                                                                                Candidate no: 3147

                                                                                                                  Centre no: 91036


In this coursework, I will be dealing with an unfamiliar task. I have to get around different words and find alternative variations for them. I must try to get different letter words, so that I can apply my knowledge and understanding to a broader scope. I have to use a range of mathematical formulas and use different mathematical techniques to explain this formula and how I got there. In the end I have to give all my mathematical formulas and give a conclusion on what I have discovered.

I can start off by finding out alternatives for some simple four-letter words.

LUCY- Different arrangements for this word:

  1. UCLY
  2. UCYL
  3. UYLC
  4. UYCL
  5. ULYC
  6. ULCY
  7. CYUL
  8. CYLU
  9. CULY
  10. CULY
  11. CLUY
  12. CLYU
  13. YCUL
  14. YLCU
  15. YULC
  16. YCLU
  17. YUCL
  18. YLUC
  19. LUCY
  20. LYCU
  21. LCUY
  22. LCYU
  23. LYUC
  24. LUYC

These were 24 different arrangements for a four-letter word, with no similar letters.

...read more.



We have investigated the three, four and five letter words. From this we can deduce the different variations for different letter words.

Now we can move onto investigating four and five letter-words with similar letters.

EMMA- This is a four-letter word with two same letters.

  1. EMMA
  2. EMAMimage00.png
  3. EAMM
  4. MEMA
  5. MAME
  6. MMEA
  7. MMAE
  8. AMME
  9. AMEM
  10. AEMM
  11. MAEM
  12. MEAM

EM1M2A - This is the same spelling as the one above, however the two M’s have been given numbers.

  1. EM1M2A
  2. EM2M1A
  3. EAM2M1
  4. EAM1M2
  5. EM1AM2
  6. EM2AM1
  7. AEM1M2
  8. AEM2M1
  9. A M2EM1
  10. A M1EM2
  11. A M2M1E
  12. AE M1M2
  13. M2AEM1
  14. M2EAM1
  15. M2 M1 AE
  16. M2 M1 E A
  17. M2 EM1A
  18. M2A EM1
  19. M1M2A E
  20. M1M2 E A
  21. M1 A M2 E
  22. M1 E M2A
  23. M1 A E M2
  24. M1 E A M2

Both EM1M2A andEMMA have the same spelling except for the subscripts. However the one without the subscripts has 12 variations while the one with the subscripts has 24.

...read more.



While discovering the formula, I went through many formulas, which were wrong. I’m going to state the formulas and show how I proved them wrong. While discovering the formula for letters with two letters, which are repeated, I thought we add the factorials in the denominator instead of multiplying it. However this was proved wrong when I tried this word:


  1. PPPQQ
  2. PPQQP
  3. PQQPP
  4. PQPPQ
  5. PQPQP
  6. PPQPP
  7. QQPPP
  8. QPPPQ
  9. QPPQP
  10. QPQPP

If the denominators were being added, I would have had 15 variations.

I also started on finding a few big words variations but decided not to carry on, e.g. Mississippi. I realized it was a tedious process and there was no point in finding it out.

These are a few other words for which I calculated variations:


No of variations









...read more.

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