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# Emma's Dilemma

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Introduction

AMDG        Maths Coursework        18/06/01

Emma’s Dilemma

Arrangements for Emma:

 emma emam eamm mmae mmea meam mema mame maem amme amem atmm

There are 12 different arrangements for Emma’s name.

Arrangements for Lucy:

 lucy ucyl cylu ycul luyc ucly cyul yclu lcuy ulcy culy yulc lcyu ulyc cuyl yucl lyuc uycl clyu ylcu lycu uylc cluy yluc

There are 24 different arrangements for Lucy’s name: Double the amount of Emma’s name.  I have noticed that with Lucy there are 6 possibilities beginning with each different letter. For example there are 6 arrangements with Lucy beginning with L, and 6 beginning with u and so on. 6 X 4 (the amount of letters) gives 24.

My strategy for working out the arrangements was using systematic patterns. For Emma’s name, I started off with the letter E and once I had found all the different arrangements beginning wit E, I moved on to M and then A.  I used the same principle for Lucy’s name.

Middle

Akmr

Amkr

Amrk

Rmak

Rmka

Rkma

Rkam

Ramk

Rakm

Kmar

Kmra

Krma

Kram

Karm

kamr

There are 24 different arrangements for Mark’s name.

As I have found that there were 24 arrangements for a 4 letter word with all different letters and that there were 6 ways beginning with one of the letter, I predict that there will be 120 arrangements for JAMES, 24 for j, 24 for a, 24 for m; etc. 120 divided by 5 (number of letters) equals 24.  Previously in the 4-letter word, 24 divided by 4 equals 6, the number of possibilities there were for each letter.

My prediction was correct as I have calculated the answer and it matches my prediction.

For Aaron, where

Conclusion

This is 4 letters with two the same

 xxyy Xyxy Yxxy Xyyx Yyxx yxyx

There are 6 arrangements

What if I had xxxyy?

x=a, y=b

 aaabb aabab aabba ababa abaab abbaa bbaaa baaab babaa baaba

There are 10 different arrangements for this instance.

What if I had an arrangement of xxxxy?

x=a, y=b

 aaaab aaaba aabaa abaaa baaaa

There are 5 different arrangements for this instance.

If I go back to xxxyy; there are 3 x’s and 2 y’s in a total of 5. As each letter has its own number of arrangements i.e. there were 5 beginning with x, and 5 beginning with y, I think that factorial has to be used again. Also in a 5-letter word there are 120 arrangements and 24 arrangements (120 divided by 5) for each letter. I came up with; the number of total letters factorial, divided by the number of x’s, y’s etc factorised and multiplied.

Formula equals n=!

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