EMMA’s Dilemma - page 2
Keywords: EMMA’s Dilemma
By Grant867 on 07/12/2008
Level: GCSE Key Stage 4 (Years 10-11)
Page Number: 2 of 5 pages: 1 2 3 4 5
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 n= number of figures
a= number of arrangements
the formular is a=ni/2
Let’s confirm the formular:
2 fig with 2 same number formular: 2/2=1 it works
3 (1*2*3)/2=3 it works
4 (1*2*3*4)/2=12 it works
Formular is confirmed
What about if 3 numbers are the same
let’s try 333
only on arrangement
Try 3331
3331
3313 ------ 4 arrangements
3133
1333
Try 33312
33312 31233 12333
33321 31323 13233 ---4 arrangements
33123 31332 ----12 arrangements 13323
33132 32331 13332
33231 32313
33213 32133
21333
23133----4 arrangements
23313
23331
Total arrangements are 4*5=20
Let’s try 6 fig with 3 same number
333124 332134
333142 332143
333214 334321
333241 332314
333412 332341 -----24 arrangements
333421 332413
331234 332431
331243 334123
331324 334132
331342 334213
331423 334231
331432 334312
312334 321334
312343 so on ----12 arrangements
312433
313234
313243 34-------
313324 --- 12 arrangements so on -----12 arrangements
313342
313423
313432
314233
314323
314332
123334 133324 2----
123343 133342 so on --------20 arrangements
123433 133234
124333 133243
124332 133423 --- 20 arrangements 4----
134323 133432 so on ---------20 arrangements
134233 142333
132334 143233
132343 143323
132433 143332
Total arrangement for 6 figure with 3 same number is 120, 20*6
Let’s see the construction:
3 fig with 3same number 1 arrangement
4 1*4
5 1*4*5
6 1*4*5*6
Can you see the pattern?
so the formula for three sames numbers of a number is:
a= ni/6
let’s review the formula:
formula for different number:
a=ni
formula for 2 same number:
a=ni/2
formular for 3 same number:
a=ni/6
Let’s put
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