Emma’s Dilemma.
Emma's Dilemma. Introduction The aim of this coursework is to investigate into the problem of different arrangements of a set of letters. By using the given guidelines and by widening the breadth of the operation I aim to find a rule or formulae that can represent and explain the situation. The basic parameters of the project are as follows: "Emma is playing with arrangements of the letters of her name. One arrangement is EMMA, another is MEAM, and another is AEMM. Investigate the number of different arrangements of the letters of Emma's name." This is the first task, which is then followed up by: "Emma has a friend named Lucy. Investigate the number of different arrangements of Lucy's name." After these two tasks I can then lead the investigation in my own direction: "Choose some different names. Investigate the number of different arrangements of the letters of the names you have chosen." An extension to the investigation is as follows: "A number of Xs and a number of Ys are written in a row such as 'XX . . . XXYY . . . Y . .' Investigate the number of different arrangements of the letters." The last section will allow me even more freedom to change variables to find more complex solutions covering a greater variety of situations. Finding The mathematical term for this type of function where a set of values is rearranged to give the maximum number of
Emma's Dilemma Question One: Investigate the number of different arrangements of the letters
Emma's Dilemma Question One: Investigate the number of different arrangements of the letters of Emma's name. Question Two: Emma has a friend named Lucy, Investigate the number of different arrangements of the letters of Lucy's name. Question Three: Choose some different names. Investigate the number of different arrangements of the letters of names you have chosen. Question Four: A number of X's and a number of Y's are written in a row such as XX.........XXYY.........Y Investigate the number of different arrangements of the letters. Question One: Investigate the number of different arrangements of the letters of EMMA's name. Answer: In order for me to answer this question, I will write down all of the different arrangements for the letters of Emma's name. This will allow me to get the total number of arrangements, which will help me to find a rule in the latter questions in this piece of coursework. Below are all of the different arrangements of the letters of Emma's name: EMMA EMAM EAMM MEMA Total: MEAM 12 MAEM MAME MMEA MMAE AMME AMEM AEMM From my results, I can see that there are 12 different combinations of the letters from Emma's name, if all of the letters are used, and only once. I have set out the letters in an ordered fashion, so that it is easier to find all of the combinations, and to only do a combination once. Therefore, my
Emma's Dilemma
GCSE Mathematics Coursework Emma's Dilemma In my investigation I am going to investigate the number of different arrangements of letters for names and words and try to find a formula that can be used to predict this. For example: TOM is one arrangement and OTM is another arrangement First, I am going to investigate the number of different arrangements of letters for the name LUCY (a 4-letter name, where all the letters are different). LUCY ULCY CLUY YLUC LUYC ULYC CLYU YLCU LCUY UCLY CULY YULC LCYU UCYL CUYL YUCL LYUC UYLC CYLU YCLU LYCU UYCL CYUL YCUL There are 4 different letters and 24 different arrangements. Once I have investigated the number of different arrangements for one 4-letter name/word where all the letters are different, I do not need to try any more. If I tried the name DAVE for example, there would still be 24 different arrangements. I could substitute the L in LUCY for the D in DAVE, the U for A, the C for V, and the Y for E; and would therefore end up with the same result. The same is true for names/words with 3 letters or 5 letters, etc. As long as the number of letters and the number of different letters are the same, the number of different arrangements will be the same. Now I will investigate a 3-letter name where all the letters are different. SAM ASM MSA SMA AMS MAS There are 6 different arrangements. Now I
I have been given a problem entitled 'Emma's Dilemma' and I was given the following information: 'Emma and Lucy are playing with arrangements of their names
Emma's Dilemma I have been given a problem entitled 'Emma's Dilemma' and I was given the following information: 'Emma and Lucy are playing with arrangements of their names. One arrangement of Lucy is: L U C Y A different arrangement is: Y L C U Part 1: Investigate the number of different arrangements of the letters of Lucy's name. Part 2: Investigate the number of different arrangements of the letters of Emma's name. Part 3: Investigate the number of different arrangements of various groups of letters.' So basically I had to investigate how the number of permutations for words alters depending on the letters that make up that word, starting off with looking at Lucy and then comparing it to Emma, and after that extending it to look at any word with various letters. I aim to produce a formula to produce the number of permutations. Part 1- Investigating Lucy: The name Lucy comprises of four different letters that we can quite simply rearrange to produce a number of arrangements. Firstly I worked out the number of arrangements that I could get from the letters of 'Lucy': LUCY LUYC LYUC LYCU LCUY LCYU ULCY ULYC UCLY UCYL UYLC UYCL CLUY CLYU CYLU CYUL CUYL YLUC YLCU YUCL YULC YCUL YCLU I found that the name Lucy can be arranged 24 different times as shown above. This would work for any four letter word as long as all the letters were different.
I am doing an investigation into words and their number of combinations. I will find formulae and work out the number of combinations for the words.
Introduction I am doing an investigation into words and their number of combinations. I will find formulae and work out the number of combinations for the words. Originally the task was to find combinations for the word EMMA and several other names, but I decided to look at sequences of letters from the alphabet, which makes it easier to monitor and control. I will start by looking at words with no letters the same, and find a formula for that, beginning with words that are 1 letter long and carrying on to letters that are 5 letters long. E.g. ABCD Next I will look at words that have two letters the same e.g ABBC, using the same method as before of starting with 2 letter words and expanding the amount of letters to 5 or 6 letter words. By doing this I am expanding the investigation and gaining more knowledge of the pattern of formulae. Throughout the investigation I will use 3 algebraic expressions. n =number of letters in total within the word. c= number of combinations found in total n!=n factorial Investigation I am going to investigate the different ways in which letters can be organised and find formulas for all of them. Letters that are all different letter 2 letters 3 letters 4 letters A AB ABC ABCD=6 BA ACB ABDC=6 Total=1 ADBC=6 Total=2 CBA DABC=6 CAB Total= 24 BAC BCA Total=6 I will now try and find a formula for this
Investigating the arrangements of letters in words.
Introduction I am doing an investigation into words and the number of ways the words can be written. I will find formulae and work out the number of combinations for the words. Originally the task was to find combinations for the word MATT, but I decided to look at sequences of letters from the alphabet, which makes it easier to monitor and control because letters can be written in sequence eg. A-B-C. I will start by looking at words with no letters the same because this is the simplest to work with. I will find a formula for that, beginning with words that are 1 letter long and carrying on to letters that are 5 letters long so I have a wide range of data to form a formula out of and test it . E.g. ABCD Next I will look at words that have two letters the same e.g. AABC because the name Matt has 2 T's in it. I will be using the same method as before but starting with 2 letter words and expanding the amount of letters to 5 or 6 letter words so again I will have a significantly large range of data to for a formula from. By doing this I am expanding the investigation and gaining more knowledge of the pattern of formulae. Throughout the investigation I will use 3 algebraic expressions. n =number of letters in total within the word. c= number of combinations found in total n!=n factorial Also I will be using simplified diagrams for large numbers. eg 4 letters
Emma’s Dilemma
Year 10 Maths Coursework Emma's Dilemma Emma is playing with different arrangements of her name. One arrangement is EMMA A different arrangement is MEAM Another arrangement is AEMM I investigated the total number of different arrangements of the letters of Emma's name. I worked out all the arrangements and come up with the following EMMA MAME MMAE AMME EMAM MAEM MAEM AMEM EAMM MEAM MAEM AEMM From this I can see that there are 12 possible arrangements. It was also interesting to see that when starting with each different letter in Emma's name, there was the same amount of arrangements for each. We had to account double the M's with there being two M's in Emma's name and so their was two times as many arrangements for that letter. Emma has a friend named Lucy. When investigating the number of different arrangements of the letters of Lucy's name, I came up with the following arrangements. LUCY CULY ULCY YUCL LCUY CLUY UCLY YCUL LYUC CYUL UYLC YLUC LYCU CYLU UYCL YLCU LCYU CLYU UCYL YCLU LUYC CUYL ULYC YULC The total number of different arrangements for the letters in Lucy's name is 24. It is interesting to see that even though the two names both have the same amount of letters in them, the number of arrangements of the letters comes to different totals. We can see that this is due to the double M in Emma, meaning that some
Emma's Dilemma
Emma's Dilemma Introduction Emma and Lucy are playing with arrangements of the letters of their names, on arrangement of Lucy is: LUCY Another arrangement is: YLCU I will be investigating all the various arrangements of 'Lucy' and all the arrangements possible for 'Emma' I will also investigate the various arrangements of words with different lengths and different letters, i.e. 3 lettered words, 4 lettered words, 5 lettered words, etc. I hope to then find a pattern which will then help me take my investigation further. I will split my investigation into three parts to make it more systematic and easier to understand. The first part will consist of me finding the various arrangements of 'Lucy' by listing them all. The second part will consist of me finding the various arrangements of 'Emma' which I will also do by listing. Finally the third part will consist of me trying to find out how many possible arrangements there are for words with various lengths and different letters, as I mentioned above (3 lettered words...). Part 1 I will be listing all the various possibilities for 'Lucy' Lucy Whilst listing the different possibilities I will try to be as systematic as I possibly can. LUCY ULCY CLUY YLCU LUYC ULYC CLYU YLUC LCYU UCLY CYLU YULC 24 POSSIBILITIES LCUY UCYL CYUL YUCL LYCU UYLC CUYL YCUL LYUC UYCL CULY YCLU In the effort of
Emma's Dilemma.
Emma's Dilemma In this investigation I am going to investigate the number of different arrangements of letters in a word. I will firstly see the number of arrangements of the letters of Emma's name. ARRANGEMENTS FOR EMMA ) EMMA 2) EMAM 3) EAMM 4) AEMM 5) AMEM 6) AMME 7) MMAE 8)MMEA 9) MEAM 10) MAEM 11) MAME 12) MEMA I have found that there are 12 different arrangements. In each word there are total of 4 letters and 2 of them are the same, 'M' which is repeated. Now I shall see how many different arrangements there are in the name Lucy ARRANGEMENTS FOR LUCY ) LUCY 2) LUYC 3) LYUC 4) LYCU 5) LCUY 6) LCYU 7) ULYC 8) ULCY 9) UYCL 10) UYLC 11) UCLY 12) UCYL 13) CLUY 14) CLYU 15) CYUL 16) CYLU 17) CULY 18) CUYL 19) YUCL 20) YULC 21) YLCU 22) YLUC 23) YCUL 24)YCLU There are 24 different arrangements possibilities in this arrangement of 4 letters that are all different, which is twice as many arrangements than EMMA. I shall now investigate on different words, which have different number of letters. I will start by using the word 'JO' ARRANGEMENTS FOR JO ) JO 2) OJ There are two arrangements for this 2-lettered name. I will now investigate the number of arrangements for 'JIM' which is a 3 lettered word ARRNAGEMENTS FOR JIM ) JIM 2) JMI 3) IJM 4) IMJ 5) MJI 6) MIJ There are 6 arrangements for this 3-lettered name, which is triple the number of arrangements
Emma's Dilemma
27th June 2001 Tom Pountain 10A Emma's Dilemma I investigated the number of different arrangements of four letters with no repetitions. )ABCD 2)ABDC 3)ADBC 4)ADCB 5)ACBD 6)ACDB 7)BACD 8)BCAD 9)BDCA 0)BADC 1)BDAC 2)CDBA 3)CBAD 5)CDAB 6)CADB 7)CBDA 8)DABC 9)DBCA 20)DCAB 21)DACB 22)DBAC 23)DCBA 24)DABC 25)BCDA I have found 24 different arrangements of these letters and this result is confirmed in the tree diagram. Secondly, I have investigated the number of different arrangements of four letters with one letter repeated twice. )ACBB 2)ABCB 3)BABC 4)BACB 5)BBAC 6)BBCA 7)BCAB 8)BCBA 9)CBBA 0)CABB 1)CBAB 2)ABBC I have found 12 different arrangements of the letters and this result is confirmed in the tree diagram. From these two investigations, I have worked out a method that can be used for further work: Firstly, with ABCD you rotate the last two letters, then you get ABDC. Then, with ABCD you must then rotate the last three letters and try the possibilities of ADBC, ADCB, ACDB, ACBD. Because the letter 'B' has been the first number of last three letters before, we don't do it again. We have list all arrangements with A first, so now we do B secondly: BACD, and we do same thing to it, it will like this: BACD=BADC, BADC=BDCA...BDAC...BCAD...BCDA We