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| Combinatorics question
| ARTRAG
online
msx master
Mensajes: 1802 |  Publicado: Enero 23 2006, 13:06 | Given a set of 16 values (the PSG levels actually)
I am looking for the number of different triplets of levels (X,Y,Z)
considerded without taking into account the order of the channels.
This means for example, that (0; 1; 0) and (1;0;0) counts for the
same triplet and are equivalent to (0,0,1).
Note that each value (among the 16) can occur 1,2 or 3 times in the
triplet indifferently.
That this imply that the solution is NOT 16!/3! / (16-3)!
So the question is:
How many different unordered triplets can be obtained from 16 values?
Hint
the result is higher than 608 and less than 4096
| | ricbit
msx lover
Mensajes: 116 | Publicado: Enero 23 2006, 14:58 | Let me do a precise calculation.
1. There are 16 different triplets with all three channels equal.
2. There are 16*15 different triplets with two channels equal. These triplets have 3 different permutations.
3. There are binomial(16,3) different triplets with no channels equal. These triplets have 6 different permutations.
The total should sum up to 16³, let's check:
16+16*15*3+16*15*14=16+720+3360=4096, ok.
The total number of states you want is the number of triplets, not counting the permutations, so it is:
16+16*15+16*15*14/6=16+240+560=816
| | AuroraMSX
msx master
Mensajes: 1278 | Publicado: Enero 23 2006, 16:03 | Question is whether eg (2,4,6), (0,6,6), or even (0,0,12), is significantly different from e.g. (4,4,4). I'm no sound expert, but my guess would be: no.
In that case you're just stuck with all triplets (X,X,X), (X,X,X+1) and (X,X+1,X+1), with X <- [0,15], leaving you no more than a mere 48 levels...
| | ARTRAG
online
msx master
Mensajes: 1802 | Publicado: Enero 23 2006, 16:20 | Aurora, trust me:
a) (2,4,6), (0,6,6) sound very different, the DAC is log type so you must compute the results to see it
b) my questions are aimed to model the PSG channel transitions so even if (0,0,3) and
(1,1,0) sound the same (and they sound the same, look at the PSG dac response) they
are different for my purposes, as lead to different behaviors during the transitions.
| | ARTRAG
online
msx master
Mensajes: 1802 | Publicado: Enero 23 2006, 22:36 | @ricbit
correct!! I found them:
>> Sv'
ans =
Columns 1 through 13
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 2 3 4 5 6 7 8 9 10 11 12
Columns 14 through 26
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 1 1 1
13 14 15 1 2 3 4 5 6 7 8 9 10
Columns 27 through 39
0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 2 2 2 2 2 2 2 2
11 12 13 14 15 2 3 4 5 6 7 8 9
Columns 40 through 52
0 0 0 0 0 0 0 0 0 0 0 0 0
2 2 2 2 2 2 3 3 3 3 3 3 3
10 11 12 13 14 15 3 4 5 6 7 8 9
Columns 53 through 65
0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 3 3 3 3 4 4 4 4 4 4 4
10 11 12 13 14 15 4 5 6 7 8 9 10
Columns 66 through 78
0 0 0 0 0 0 0 0 0 0 0 0 0
4 4 4 4 4 5 5 5 5 5 5 5 5
11 12 13 14 15 5 6 7 8 9 10 11 12
Columns 79 through 91
0 0 0 0 0 0 0 0 0 0 0 0 0
5 5 5 6 6 6 6 6 6 6 6 6 6
13 14 15 6 7 8 9 10 11 12 13 14 15
Columns 92 through 104
0 0 0 0 0 0 0 0 0 0 0 0 0
7 7 7 7 7 7 7 7 7 8 8 8 8
7 8 9 10 11 12 13 14 15 8 9 10 11
Columns 105 through 117
0 0 0 0 0 0 0 0 0 0 0 0 0
8 8 8 8 9 9 9 9 9 9 9 10 10
12 13 14 15 9 10 11 12 13 14 15 10 11
Columns 118 through 130
0 0 0 0 0 0 0 0 0 0 0 0 0
10 10 10 10 11 11 11 11 11 12 12 12 12
12 13 14 15 11 12 13 14 15 12 13 14 15
Columns 131 through 143
0 0 0 0 0 0 1 1 1 1 1 1 1
13 13 13 14 14 15 1 1 1 1 1 1 1
13 14 15 14 15 15 1 2 3 4 5 6 7
Columns 144 through 156
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 2 2 2 2 2
8 9 10 11 12 13 14 15 2 3 4 5 6
Columns 157 through 169
1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 3 3 3 3
7 8 9 10 11 12 13 14 15 3 4 5 6
Columns 170 through 182
1 1 1 1 1 1 1 1 1 1 1 1 1
3 3 3 3 3 3 3 3 3 4 4 4 4
7 8 9 10 11 12 13 14 15 4 5 6 7
Columns 183 through 195
1 1 1 1 1 1 1 1 1 1 1 1 1
4 4 4 4 4 4 4 4 5 5 5 5 5
8 9 10 11 12 13 14 15 5 6 7 8 9
Columns 196 through 208
1 1 1 1 1 1 1 1 1 1 1 1 1
5 5 5 5 5 5 6 6 6 6 6 6 6
10 11 12 13 14 15 6 7 8 9 10 11 12
Columns 209 through 221
1 1 1 1 1 1 1 1 1 1 1 1 1
6 6 6 7 7 7 7 7 7 7 7 7 8
13 14 15 7 8 9 10 11 12 13 14 15 8
Columns 222 through 234
1 1 1 1 1 1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 9 9 9 9 9 9
9 10 11 12 13 14 15 9 10 11 12 13 14
Columns 235 through 247
1 1 1 1 1 1 1 1 1 1 1 1 1
9 10 10 10 10 10 10 11 11 11 11 11 12
15 10 11 12 13 14 15 11 12 13 14 15 12
Columns 248 through 260
1 1 1 1 1 1 1 1 1 2 2 2 2
12 12 12 13 13 13 14 14 15 2 2 2 2
13 14 15 13 14 15 14 15 15 2 3 4 5
Columns 261 through 273
2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 3 3 3
6 7 8 9 10 11 12 13 14 15 3 4 5
Columns 274 through 286
2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 4 4 4
6 7 8 9 10 11 12 13 14 15 4 5 6
Columns 287 through 299
2 2 2 2 2 2 2 2 2 2 2 2 2
4 4 4 4 4 4 4 4 4 5 5 5 5
7 8 9 10 11 12 13 14 15 5 6 7 8
Columns 300 through 312
2 2 2 2 2 2 2 2 2 2 2 2 2
5 5 5 5 5 5 5 6 6 6 6 6 6
9 10 11 12 13 14 15 6 7 8 9 10 11
Columns 313 through 325
2 2 2 2 2 2 2 2 2 2 2 2 2
6 6 6 6 7 7 7 7 7 7 7 7 7
12 13 14 15 7 8 9 10 11 12 13 14 15
Columns 326 through 338
2 2 2 2 2 2 2 2 2 2 2 2 2
8 8 8 8 8 8 8 8 9 9 9 9 9
8 9 10 11 12 13 14 15 9 10 11 12 13
Columns 339 through 351
2 2 2 2 2 2 2 2 2 2 2 2 2
9 9 10 10 10 10 10 10 11 11 11 11 11
14 15 10 11 12 13 14 15 11 12 13 14 15
Columns 352 through 364
2 2 2 2 2 2 2 2 2 2 3 3 3
12 12 12 12 13 13 13 14 14 15 3 3 3
12 13 14 15 13 14 15 14 15 15 3 4 5
Columns 365 through 377
3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 4 4 4
6 7 8 9 10 11 12 13 14 15 4 5 6
Columns 378 through 390
3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 5 5 5 5
7 8 9 10 11 12 13 14 15 5 6 7 8
Columns 391 through 403
3 3 3 3 3 3 3 3 3 3 3 3 3
5 5 5 5 5 5 5 6 6 6 6 6 6
9 10 11 12 13 14 15 6 7 8 9 10 11
Columns 404 through 416
3 3 3 3 3 3 3 3 3 3 3 3 3
6 6 6 6 7 7 7 7 7 7 7 7 7
12 13 14 15 7 8 9 10 11 12 13 14 15
Columns 417 through 429
3 3 3 3 3 3 3 3 3 3 3 3 3
8 8 8 8 8 8 8 8 9 9 9 9 9
8 9 10 11 12 13 14 15 9 10 11 12 13
Columns 430 through 442
3 3 3 3 3 3 3 3 3 3 3 3 3
9 9 10 10 10 10 10 10 11 11 11 11 11
14 15 10 11 12 13 14 15 11 12 13 14 15
Columns 443 through 455
3 3 3 3 3 3 3 3 3 3 4 4 4
12 12 12 12 13 13 13 14 14 15 4 4 4
12 13 14 15 13 14 15 14 15 15 4 5 6
Columns 456 through 468
4 4 4 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 5 5 5 5
7 8 9 10 11 12 13 14 15 5 6 7 8
Columns 469 through 481
4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 6 6 6 6 6 6
9 10 11 12 13 14 15 6 7 8 9 10 11
Columns 482 through 494
4 4 4 4 4 4 4 4 4 4 4 4 4
6 6 6 6 7 7 7 7 7 7 7 7 7
12 13 14 15 7 8 9 10 11 12 13 14 15
Columns 495 through 507
4 4 4 4 4 4 4 4 4 4 4 4 4
8 8 8 8 8 8 8 8 9 9 9 9 9
8 9 10 11 12 13 14 15 9 10 11 12 13
Columns 508 through 520
4 4 4 4 4 4 4 4 4 4 4 4 4
9 9 10 10 10 10 10 10 11 11 11 11 11
14 15 10 11 12 13 14 15 11 12 13 14 15
Columns 521 through 533
4 4 4 4 4 4 4 4 4 4 5 5 5
12 12 12 12 13 13 13 14 14 15 5 5 5
12 13 14 15 13 14 15 14 15 15 5 6 7
Columns 534 through 546
5 5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 6 6 6 6 6
8 9 10 11 12 13 14 15 6 7 8 9 10
Columns 547 through 559
5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 7 7 7 7 7 7 7 7
11 12 13 14 15 7 8 9 10 11 12 13 14
Columns 560 through 572
5 5 5 5 5 5 5 5 5 5 5 5 5
7 8 8 8 8 8 8 8 8 9 9 9 9
15 8 9 10 11 12 13 14 15 9 10 11 12
Columns 573 through 585
5 5 5 5 5 5 5 5 5 5 5 5 5
9 9 9 10 10 10 10 10 10 11 11 11 11
13 14 15 10 11 12 13 14 15 11 12 13 14
Columns 586 through 598
5 5 5 5 5 5 5 5 5 5 5 6 6
11 12 12 12 12 13 13 13 14 14 15 6 6
15 12 13 14 15 13 14 15 14 15 15 6 7
Columns 599 through 611
6 6 6 6 6 6 6 6 6 6 6 6 6
6 6 6 6 6 6 6 6 7 7 7 7 7
8 9 10 11 12 13 14 15 7 8 9 10 11
Columns 612 through 624
6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 8 8 8 8 8 8 8 8 9
12 13 14 15 8 9 10 11 12 13 14 15 9
Columns 625 through 637
6 6 6 6 6 6 6 6 6 6 6 6 6
9 9 9 9 9 9 10 10 10 10 10 10 11
10 11 12 13 14 15 10 11 12 13 14 15 11
Columns 638 through 650
6 6 6 6 6 6 6 6 6 6 6 6 6
11 11 11 11 12 12 12 12 13 13 13 14 14
12 13 14 15 12 13 14 15 13 14 15 14 15
Columns 651 through 663
6 7 7 7 7 7 7 7 7 7 7 7 7
15 7 7 7 7 7 7 7 7 7 8 8 8
15 7 8 9 10 11 12 13 14 15 8 9 10
Columns 664 through 676
7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 9 9 9 9 9 9 9 10
11 12 13 14 15 9 10 11 12 13 14 15 10
Columns 677 through 689
7 7 7 7 7 7 7 7 7 7 7 7 7
10 10 10 10 10 11 11 11 11 11 12 12 12
11 12 13 14 15 11 12 13 14 15 12 13 14
Columns 690 through 702
7 7 7 7 7 7 7 8 8 8 8 8 8
12 13 13 13 14 14 15 8 8 8 8 8 8
15 13 14 15 14 15 15 8 9 10 11 12 13
Columns 703 through 715
8 8 8 8 8 8 8 8 8 8 8 8 8
8 8 9 9 9 9 9 9 9 10 10 10 10
14 15 9 10 11 12 13 14 15 10 11 12 13
Columns 716 through 728
8 8 8 8 8 8 8 8 8 8 8 8 8
10 10 11 11 11 11 11 12 12 12 12 13 13
14 15 11 12 13 14 15 12 13 14 15 13 14
Columns 729 through 741
8 8 8 8 9 9 9 9 9 9 9 9 9
13 14 14 15 9 9 9 9 9 9 9 10 10
15 14 15 15 9 10 11 12 13 14 15 10 11
Columns 742 through 754
9 9 9 9 9 9 9 9 9 9 9 9 9
10 10 10 10 11 11 11 11 11 12 12 12 12
12 13 14 15 11 12 13 14 15 12 13 14 15
Columns 755 through 767
9 9 9 9 9 9 10 10 10 10 10 10 10
13 13 13 14 14 15 10 10 10 10 10 10 11
13 14 15 14 15 15 10 11 12 13 14 15 11
Columns 768 through 780
10 10 10 10 10 10 10 10 10 10 10 10 10
11 11 11 11 12 12 12 12 13 13 13 14 14
12 13 14 15 12 13 14 15 13 14 15 14 15
Columns 781 through 793
10 11 11 11 11 11 11 11 11 11 11 11 11
15 11 11 11 11 11 12 12 12 12 13 13 13
15 11 12 13 14 15 12 13 14 15 13 14 15
Columns 794 through 806
11 11 11 12 12 12 12 12 12 12 12 12 12
14 14 15 12 12 12 12 13 13 13 14 14 15
14 15 15 12 13 14 15 13 14 15 14 15 15
Columns 807 through 816
13 13 13 13 13 13 14 14 14 15
13 13 13 14 14 15 14 14 15 15
13 14 15 14 15 15 14 15 15 15
| | AuroraMSX
msx master
Mensajes: 1278 | Publicado: Enero 24 2006, 09:58 | Quote:
| Aurora, trust me:
a) (2,4,6), (0,6,6) sound very different, the DAC is log type so you must compute the results to see it
|
Ah, ok. I see. And I do trust you 
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