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Murray Goodwin GCSE Maths Project - "Emma's Dilemma" Part 1: Ways of arranging EMMA's name: EMMA MEMA MMAE AMME EMAM MEAM MAEM AMEM EAMM MMEA MAME AEMM For each new beginning letter, there are 3 different possible combinations. There are a total of 12 possible different combinations. Part 2: Ways of arranging LUCY's name: LUCY ULCY CULY YUCL LUYC ULYC CUYL YULC LCUY UCLY CLUY YCUL LCYU UCYL CLYU YCLU LYUC UYLC CYLU YLUC LYCU UYCL CYUL YLCU For each new beginning letter, there are 6 different possible combinations. There are a total of 24 possible different combinations. Summary of Parts 1 and 2: EMMA and LUCY both have the same number of letters in their names, however LUCY has twice as many different letter combinations. This is because EMMA's name has a repeating letter (in this case the letter "M" is repeated) whereas the letters in LUCY's name are all different. Where a name has four letters (A1, A2, B, and C), the possible arrangements can be written as follows: A1A2BC A2A1BC BA2A1C CA2BA1 A1A2CB A2A1CB BA2CA1 CA2A1B A1BA2C A2BA1C BA1A2C CBA2A1 A1BCA2 A2BCA1 BA1CA2 CBA1A2 A1CA2B A2CA1B BCA1A2 CA1A2B A1CBA2 A2CBA1 BCA2A1 CA1BA2 This gives a total of 24 possible combinations. However, if we take the letters again, and this time remove the subscript numbers from the letter A, we can...

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