Emmas Dilemma Essay Research Paper Firstly we

Emma.s Dilemma Essay, Research Paper

Firstly we arrange EMMA.s Name. 1)EAMM

7)MAEM 2)EMAM

8)MAME 3)MEMA

9)AMME 4)MEAM

10)AEMM 5)MMEA

11)AMEM 6)MMAE

12)EMMA Secondlywe arrange lucy.s name. 1)Lucy

12)Cyul 22)Yulc 2)Luyc

13)Culy 23)Ycul 4)Lycu

14)Culy 24)Yluc 5)Lcuy

15)Cylu 25)Ucyl 6)Lcyu

16)Clyu 7)Ulcy

17)Cuyl 8)Ucly

18)Yluc 9)Uycl

19)Yucl 10)Ulyc

20)Yclu 11)Uylc

21)Ylcu From these 2 investigation I worked out a method: Step1: 1234—Do the last two number first then you get 1243.

1243—Do the last three numbers and try the possibility. 1423. 1432.

1342.

1324, because the number 2 has been the first number of last three

numbers, so

we don.t do it again. Step2:

we have list all arrangements of 1 go front, so we do 2 go front. 2134

and

we do same thing to it, it will like this:

2134—2143, 2143—2431,2413,2314,2341 Step3:

We have finished 2 go first, then let.s do 3 go ahead.

3124—3142, 3142—3241,3214,3412,3421 Step4:

We have finished 3 go ahead, then try 4

4123—4132, 4132—4231,4213,4312,4321 We have list all arrangement of 1234, use this method we can arrange the

number which has 5 figures or more. We are trying to work out a formula which can calculate the number of

arrangement when we look at a number. Let.s list all the arrangment for 1234: 1234

4123 1243

4132 1324

— 6

arrangment

4231 —- 6 arrangement 1342

4213 1432

4321 1423

4312 2134

3124 2143

3143 2341

—- 6 arrangements 3241

— 6 arrangements 2314

3214 2413

3412 2431

3421 So if we time 6 by 4, we would get 24, and we

can get the total arrangements of 24. Let.s try 5 figures: 12345

13245 12354

13254 12435

— 6 arrangements

13452 — 6 arrangements 12453

13425 12534

13524 12543

13542 14235

15432 14253

15423 14352

— 6 arrangements

15324 —- 6 arrangements 14523

15342 14532

15243 14325

15234 Don.t you notice the arrangements of last 4

numbers added up, it equal 24, so if we time 24 by 5, and get 120, and 120 is

the total of arrangements. Carry on, if a number has 6 figure, then the

total of arrangement should be 120 times by 6, and get 720, and 720 is the

total of arrangements. Carry on, if a number has 7 fugure, then the

total of arrangement should be 720 times by 7, and get 5040, the total of

arrangement is 5040. This is my prediction, let.s work it out a

formula, and confirm it. 3

figure with different

number it has 6

arrangements 4

4*6 5

4*6*5 6

4*6*5*6 7

4*6*5*6*7 8

4*6*5*6*7*8 so on We can rewrite it as: 1

fig 1 2

fig 1*2 3

fig 1*2*3 4

fig 1*2*3*4 5

fig 1*2*3*4*5 6

fig 1*2*3*4*5*6 so on There.s a simbol for the frequancy above,

that.s I.For example: 1*2 = 2i 1*2*3 = 3i 1*2*3*4 = 4i so on So if n represent the number of figures of a

number, then it has arrangements of ni. The

formula: NI NI: Can be caculated on caculator. Process: pres key N (the number of figure),

then press key I, then you would get the arrangements. Let.s confirm this formula, if a nuber has 1fig

it has 1 arrangements formular: 1*1=1 It works 2fig

it has 2 arrangements formular: 1*2=2 It works 3fig

it has 6 arrangements formular: 1*2*3=6 It works 4fig

it has 24 arrangements formular: 1*2*3*4=24 It works Formula is confirmed What about if a number has two same figure For

example: 223, 334 Let.s try to work out the frequency of them 223 can be arranged as 232, 322 only 3 arrangements try 4 figures with 2 same numbers 1223 arranged as: 1223

2123

3122 1232

—- 3 arrangements

2132

3212 — 3 arrrangements 1322

2213 —- 6 arrangements 3221

2231

2312

2321 total arrangement is 12 Try 5 fig: 42213

12234 42231

12243 42123

12324 42132

12342 42321

12423 42312

——– 12 arrangements

12432 ——– 12 arrangements 41223

13224 41232

13242 41322

13422 43122

14223 43212

14232 43221

14322 21234

23124

31224 21243

23142

31242 21324

23214

31422 21342

23241

32124 21423

23421

32142 21421

23412 —–24 arrangments

32214 ——- 12 arrangements 22134

24123

32241 22143

24132

32412 22314

24231

32421 22341

24213

34122 22413

24312

34212 22431

24321

34221 so the total arrangements are 12*5=60 We have found the frequency 2

figure with 2 same

number

1arrangements 3

1*3 4

1*3*4 5

1*3*4*5 Let.s work out the formular: if

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