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7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
May 20th, 2019 at 5:25:31 PM permalink
Quote: ThatDonGuy

How to get such a count without having to go through all 280 products, I don't know, and presumably, that's why things like Mathematica exist.

what you say is so true.
I looked for the original pari/gp text file that is created from that formula and I do not have it and do not want to make it. I think it was very large and a program as notepad++ only could open it and make things happen.
added: I made that file and it is only 1.9MB in a text file and took less than 1 second to make.

Here is what I do have for x^20 (draw 20)
y^sum is the sum and the coefficient is the number of ways that sum can occur.
this can be compared to by one who really wants to do this

This concept is about partitions

here is 2 numbers and 3 numbers expanded and it should make sense.
gp > a=prod(i=1, 2, 1 + x*y^i);
gp > b=prod(i=1, 3, 1 + x*y^i);
gp > a
%3 = y^3*x^2 + (y^2 + y)*x + 1
gp > b
%4 = y^6*x^3 + (y^5 + y^4 + y^3)*x^2 + (y^3 + y^2 + y)*x + 1

a=prod(i=1, 2, 1 + x*y^i); means we only have 2 numbers (1 and 2)
y^3*x^2 + (y^2 + y)*x + 1 means drawing 2 numbers (x^2) we have only 1 sum and that is y^3 = 3 and only 1 way to get it (1+2)
(y^2 + y)*x means we draw 1 number (x) and the sums are 2 and 1 and 1 way to get each

same concept with numbers {1,2,3}
x^20
1*y^1410
1*y^1409
2*y^1408
3*y^1407
5*y^1406
7*y^1405
11*y^1404
15*y^1403
22*y^1402
30*y^1401
42*y^1400
56*y^1399
77*y^1398
101*y^1397
135*y^1396
176*y^1395
231*y^1394
297*y^1393
385*y^1392
490*y^1391
627*y^1390
791*y^1389
1000*y^1388
1251*y^1387
1568*y^1386
1946*y^1385
2417*y^1384
2980*y^1383
3673*y^1382
4498*y^1381
5507*y^1380
6703*y^1379
8154*y^1378
9871*y^1377
11937*y^1376
14375*y^1375
17293*y^1374
20722*y^1373
24803*y^1372
29588*y^1371
35251*y^1370
41869*y^1369
49668*y^1368
58754*y^1367
69414*y^1366
81801*y^1365
96271*y^1364
113039*y^1363
132559*y^1362
155112*y^1361
181274*y^1360
211428*y^1359
246288*y^1358
286364*y^1357
332557*y^1356
385528*y^1355
446405*y^1354
516054*y^1353
595872*y^1352
686983*y^1351
791131*y^1350
909741*y^1349
1044984*y^1348
1198689*y^1347
1373524*y^1346
1571812*y^1345
1796855*y^1344
2051569*y^1343
2340024*y^1342
2665885*y^1341
3034135*y^1340
3449359*y^1339
3917670*y^1338
4444748*y^1337
5038070*y^1336
5704686*y^1335
6453684*y^1334
7293767*y^1333
8236001*y^1332
9291060*y^1331
10472375*y^1330
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13269250*y^1328
14917024*y^1327
16755957*y^1326
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23630664*y^1323
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33091578*y^1320
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51303148*y^1316
57143201*y^1315
63604540*y^1314
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4691744778*y^1268
5093097335*y^1267
5526511602*y^1266
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2563361846130743*y^638
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2199737298000372*y^631
2151017310973098*y^630
2103089813703016*y^629
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2009591422664545*y^627
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1747542887215360*y^621
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1549571094266055*y^616
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1475373688210195*y^614
1439316353708705*y^613
1403943585301204*y^612
1369248008480895*y^611
1335222184292444*y^610
1301858588751637*y^609
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1205665050808215*y^606
1174873945708909*y^605
1144706586113291*y^604
1115155123030642*y^603
1086211682173956*y^602
1057868345195045*y^601
1030117177412699*y^600
1002950209453776*y^599
976359464227661*y^598
950336938960611*y^597
924874631401209*y^596
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875598600626154*y^594
851768846820577*y^593
828467257050350*y^592
805685826594439*y^591
783416573752126*y^590
761651523370470*y^589
740382730055707*y^588
719602262037791*y^587
699302223666421*y^586
679474739604064*y^585
660111976597920*y^584
641206128010489*y^583
622749434876858*y^582
604734170747184*y^581
587152662058258*y^580
569997273279789*y^579
553260426593322*y^578
536934587353256*y^577
521012283085036*y^576
505486089237058*y^575
490348647526558*y^574
475592651976889*y^573
461210866607609*y^572
447196111767398*y^571
433541281180913*y^570
420239328556562*y^569
407283284019903*y^568
394666240991089*y^567
382381372002154*y^566
370421915855964*y^565
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347452592155562*y^563
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325705719979621*y^561
315274621737283*y^560
305129994336046*y^559
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285675384124940*y^557
276353217726061*y^556
267293156421037*y^555
258489317615563*y^554
249935896266269*y^553
241627176819381*y^552
233557522163930*y^551
225721385084601*y^550
218113297456569*y^549
210727881214639*y^548
203559837800120*y^547
196603958660512*y^546
189855114934011*y^545
183308267507535*y^544
176958456936318*y^543
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159042951225485*y^540
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142730412563985*y^537
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132686251275763*y^535
127900559773363*y^534
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118781314944316*y^532
114439747308027*y^531
110238340304328*y^530
106173282459017*y^529
102240843924128*y^528
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94759279010024*y^526
91203071830294*y^525
87765320677722*y^524
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78129624219915*y^521
75132884036461*y^520
72238553020681*y^519
69443636407019*y^518
66745205168528*y^517
64140401221500*y^516
61626430488492*y^515
59200567856702*y^514
56860150445667*y^513
54602582321801*y^512
52425327976532*y^511
50325916807381*y^510
48301936795528*y^509
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31722346290416*y^499
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3216909849508*y^451
3051434096031*y^450
2893821211946*y^449
2743731467669*y^448
2600837554495*y^447
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2212249424868*y^444
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1681489788194*y^439
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1422278292091*y^436
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712274211009*y^424
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528282584833*y^419
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293350487*y^321
266062168*y^320
241171101*y^319
218482569*y^318
197809729*y^317
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161856093*y^315
146276124*y^314
132112328*y^313
119246146*y^312
107563103*y^311
96963169*y^310
87349501*y^309
78637842*y^308
70746444*y^307
63604540*y^306
57143201*y^305
51303148*y^304
46026431*y^303
41263462*y^302
36965620*y^301
33091578*y^300
29600587*y^299
26458297*y^298
23630664*y^297
21089176*y^296
18805464*y^295
16755957*y^294
14917024*y^293
13269250*y^292
11793028*y^291
10472375*y^290
9291060*y^289
8236001*y^288
7293767*y^287
6453684*y^286
5704686*y^285
5038070*y^284
4444748*y^283
3917670*y^282
3449359*y^281
3034135*y^280
2665885*y^279
2340024*y^278
2051569*y^277
1796855*y^276
1571812*y^275
1373524*y^274
1198689*y^273
1044984*y^272
909741*y^271
791131*y^270
686983*y^269
595872*y^268
516054*y^267
446405*y^266
385528*y^265
332557*y^264
286364*y^263
246288*y^262
211428*y^261
181274*y^260
155112*y^259
132559*y^258
113039*y^257
96271*y^256
81801*y^255
69414*y^254
58754*y^253
49668*y^252
41869*y^251
35251*y^250
29588*y^249
24803*y^248
20722*y^247
17293*y^246
14375*y^245
11937*y^244
9871*y^243
8154*y^242
6703*y^241
5507*y^240
4498*y^239
3673*y^238
2980*y^237
2417*y^236
1946*y^235
1568*y^234
1251*y^233
1000*y^232
791*y^231
627*y^230
490*y^229
385*y^228
297*y^227
231*y^226
176*y^225
135*y^224
101*y^223
77*y^222
56*y^221
42*y^220
30*y^219
22*y^218
15*y^217
11*y^216
7*y^215
5*y^214
3*y^213
2*y^212
1*y^211
1*y^210
the distribution is symmetrical. 810 being the center value (well known to Keno folks)
Last edited by: 7craps on May 20, 2019
winsome johnny (not Win some johnny)
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1324
  • Posts: 21759
May 20th, 2019 at 8:08:11 PM permalink
There is blood all over the wall from me trying to make sense of the above post. Please keep in mind I don't have Mathematica and I'm looking for a way to express the combinations in ordinary mathematical language.
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
May 20th, 2019 at 8:37:37 PM permalink
Quote: Wizard

Please keep in mind I don't have Mathematica

LOL
Mathematica it is not required.
a program that can expand a product function accurately is all that is needed. That is why I use pari/gp as it is free (windows 64 version) and the learning curve is not steep at all if you already know another language.
Quote: Wizard

and I'm looking for a way to express the combinations in ordinary mathematical language.

Don showed why the formula works and it is actually very simple once you understand the concept of a generating function.

I guess I just got lucky as the light switch went on for me. On a scale of 1 to 10 , my math skills are about a 2
your math skills may be way different from mine, but this method of using a generating function is very basic, as generating functions go.

IF you can understand how a GF works with 2d6 and dice sums, you can get this product function.

(x+x^2+x^3+x^4+x^5+x^6)^2. most know how to expand this the long way. we learned this way back in school.
now we have computers that do it for us in 0 ms

you will get it in no time.
winsome johnny (not Win some johnny)
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 94
  • Posts: 4189
May 20th, 2019 at 8:58:42 PM permalink
Quote: ThatDonGuy

How to get such a count without having to go through all 280 products, I don't know, and presumably, that's why things like Mathematica exist.


I did find this, however, which explains the problem and a way to find the solution
ZPP
ZPP
Joined: Feb 7, 2010
  • Threads: 2
  • Posts: 31
May 21st, 2019 at 12:12:49 AM permalink
Using a computer, this can be calculated using a recurrence:
Let A[n][k][m] be the number of ways to choose k numbers from {1..n} that sum to m.
Suppose we are trying to calculate A[n][k][m] and have already calculated all values for A[n-1][k][m].
We add together the number of ways that do not include n and the number of ways that do include n. This is exhaustive and disjoint.
1. The number of ways that do not include n is A[n-1][k][m], the number of ways to choose k numbers from {1..n-1} that sum to m.
2. If m>=n, the number of ways that do include n is A[n-1][k-1][m-n], the number of ways to choose k-1 numbers from {1..n-1} that sum to m-n. (If m<n, the number of ways to sum to m that do include n is 0.)

This is very fast: using bounds for n, k, and m of 80, 20, and 80*20 (which is overkill, could be 1410), the number of elements to calculate is 80*20*80*20=2,560,000, and each element is at most two array lookups and an addition. It takes less than 10 milliseconds on my laptop.

Here is a C program that does this calculation:
#include <stdio.h>
#include <stdint.h>

#define MAXK 20
#define MAXN 80
#define MAXM (MAXK*MAXN)

// Since 80 choose 20 < 2^64, we can get exact counts with 64-bit ints
// (but note that 80 choose 22 >= 2^64)
typedef uint64_t num_t;
// Just index by size_t
typedef size_t index_t;

// a[n][k][m] is the number of ways to choose k numbers from {1..n} that sum to m
num_t a[MAXN+1][MAXK+1][MAXM+1];
// (Note that because we only reference a[n-1] when calculating a[n],
// we only need two (current and previous) 2-dimensional arrays of size
// MAXK+1 by MAXM+1, not a 3-dimensional array, but nowadays the
// memory needed is nothing.)

int main(void)
{
for (index_t n = 0; n <= MAXN; n++) {
// For any set (including the empty set for n==0), there is 1
// way to choose k==0 numbers and get a sum of 0 and no other
// sum is possible by choosing 0 numbers.
a[n][0][0] = 1;
// Now calculate for choosing more than k>0 numbers.
// We iterate k from 1 to min(n,MAXK).
// Note that therefore we skip this loop if n==0.
index_t maxk = (n < MAXK ? n : MAXK);
for (index_t k = 1; k <= maxk; k++) {
// maxm could be tighter, but this is already fast enough.
index_t maxm = n * k;
for (index_t m = 0; m <= maxm; m++) {
// Calculate a[n][k][m] by adding the number of ways
// that do not include n in the sum and the number of
// ways that do include n in the sum, both of which
// are already calculated in the array.
//
// The number of ways that do not include n is the
// number of ways to choose k numbers from {1..n-1}
// that sum to m.
num_t c = a[n-1][k][m];
// The number of ways that do include n is the number of
// ways to choose k-1 numbers from {1..n-1} that sum
// to m-n, or 0 if m<n.
if (n <= m)
c += a[n-1][k-1][m-n];
a[n][k][m] = c;
}
}
}
// Print results, which are a table listing M and the number of
// ways of choosing MAXK numbers from {1..MAXN} that sum to M
num_t total = 0;
num_t *p = a[MAXN][MAXK];
for (index_t m = 0; m <= MAXM; m++) {
total += p[m];
if (p[m] != 0)
printf("%4zu %19llu\n", m, p[m]);
}
// Total number of ways can be checked to equal MAXN choose MAXK
printf("Sum: %19llu\n", total);
return 0;
}


Here are the results I get:
 210                   1
211 1
212 2
213 3
214 5
215 7
216 11
217 15
218 22
219 30
220 42
221 56
222 77
223 101
224 135
225 176
226 231
227 297
228 385
229 490
230 627
231 791
232 1000
233 1251
234 1568
235 1946
236 2417
237 2980
238 3673
239 4498
240 5507
241 6703
242 8154
243 9871
244 11937
245 14375
246 17293
247 20722
248 24803
249 29588
250 35251
251 41869
252 49668
253 58754
254 69414
255 81801
256 96271
257 113039
258 132559
259 155112
260 181274
261 211428
262 246288
263 286364
264 332557
265 385528
266 446405
267 516054
268 595872
269 686983
270 791131
271 909741
272 1044984
273 1198689
274 1373524
275 1571812
276 1796855
277 2051569
278 2340024
279 2665885
280 3034135
281 3449359
282 3917670
283 4444748
284 5038070
285 5704686
286 6453684
287 7293767
288 8236001
289 9291060
290 10472375
291 11793028
292 13269250
293 14917024
294 16755957
295 18805464
296 21089176
297 23630664
298 26458297
299 29600587
300 33091578
301 36965620
302 41263462
303 46026431
304 51303148
305 57143201
306 63604540
307 70746444
308 78637842
309 87349501
310 96963169
311 107563103
312 119246146
313 132112328
314 146276124
315 161856093
316 178987225
317 197809729
318 218482569
319 241171101
320 266062168
321 293350487
322 323255164
323 356005000
324 391856581
325 431078138
326 473969587
327 520844871
328 572053851
329 627963266
330 688980643
331 755533468
332 828095629
333 907164667
334 993290571
335 1087051347
336 1189084411
337 1300059954
338 1420715489
339 1551826889
340 1694245990
341 1848869496
342 2016679881
343 2198711618
344 2396096082
345 2610024936
346 2841798846
347 3092788314
348 3364486581
349 3658467191
350 3976441876
351 4320214692
352 4691744778
353 5093097335
354 5526511602
355 5994347965
356 6499162145
357 7043648186
358 7630718671
359 8263443950
360 8945138936
361 9679297643
362 10469687646
363 11320279729
364 12235349889
365 13219404469
366 14277290236
367 15414114032
368 16635362300
369 17946814901
370 19354673930
371 20865472184
372 22486212019
373 24224267278
374 26087533643
375 28084323773
376 30223529082
377 32514508546
378 34967262684
379 37592314751
380 40400898786
381 43404832757
382 46616720518
383 50049817550
384 53718247833
385 57636860670
386 61821464449
387 66288674187
388 71056162189
389 76142496873
390 81567411419
391 87351632249
392 93517168095
393 100087127594
394 107086028628
395 114539605867
396 122475141756
397 130921262045
398 139908291043
399 149468034620
400 159634159836
401 170441966142
402 181928790813
403 194133766301
404 207098254570
405 220865589927
406 235481542190
407 250994045973
408 267453694512
409 284913452879
410 303429185857
411 323059354425
412 343865577844
413 365912314318
414 389267459099
415 414002006749
416 440190689221
417 467911618783
418 497246966079
419 528282584833
420 561108732221
421 595819672189
422 632514442353
423 671296435063
424 712274211009
425 755561059108
426 801275859118
427 849542617012
428 900491381529
429 954257753733
430 1010983857544
431 1070817824773
432 1133914822434
433 1200436509276
434 1270552124993
435 1344437917067
436 1422278292091
437 1504265214046
438 1590599420536
439 1681489788194
440 1777154619819
441 1877820975168
442 1983726028916
443 2095116368246
444 2212249424868
445 2335392734382
446 2464825448513
447 2600837554495
448 2743731467669
449 2893821211946
450 3051434096031
451 3216909849508
452 3390602389686
453 3572878910940
454 3764121741811
455 3964727389027
456 4175108488550
457 4395692797816
458 4626925247931
459 4869266881582
460 5123197006289
461 5389212078927
462 5667827963566
463 5959578755669
464 6265019152487
465 6584723213704
466 6919286843610
467 7269326489361
468 7635481738958
469 8018413948848
470 8418808965914
471 8837375680898
472 9274848873719
473 9731987693363
474 10209578630140
475 10708433913262
476 11229394619008
477 11773328981483
478 12341135635235
479 12933741839899
480 13552106861442
481 14197220100303
482 14870104620270
483 15571815175325
484 16303441884554
485 17066108157849
486 17860974520443
487 18689236426307
488 19552128242181
489 20450920942555
490 21386926250737
491 22361494215565
492 23376017513498
493 24431928894154
494 25530705653070
495 26673866939713
496 27862978398196
497 29099649337087
498 30385537541434
499 31722346290416
500 33111829351342
501 34555787837266
502 36056075379039
503 37614594822327
504 39233303581272
505 40914210159126
506 42659379693854
507 44470930294083
508 46351038772509
509 48301936795528
510 50325916807381
511 52425327976532
512 54602582321801
513 56860150445667
514 59200567856702
515 61626430488492
516 64140401221500
517 66745205168528
518 69443636407019
519 72238553020681
520 75132884036461
521 78129624219915
522 81231841219122
523 84442670093739
524 87765320677722
525 91203071830294
526 94759279010024
527 98437368245632
528 102240843924128
529 106173282459017
530 110238340304328
531 114439747308027
532 118781314944316
533 123266929349329
534 127900559773363
535 132686251275763
536 137628133409981
537 142730412563985
538 147997380868998
539 153433408182971
540 159042951225485
541 164830545178892
542 170800813062207
543 176958456936318
544 183308267507535
545 189855114934011
546 196603958660512
547 203559837800120
548 210727881214639
549 218113297456569
550 225721385084601
551 233557522163930
552 241627176819381
553 249935896266269
554 258489317615563
555 267293156421037
556 276353217726061
557 285675384124940
558 295265627053394
559 305129994336046
560 315274621737283
561 325705719979621
562 336429586543371
563 347452592155562
564 358781192841696
565 370421915855964
566 382381372002154
567 394666240991089
568 407283284019903
569 420239328556562
570 433541281180913
571 447196111767398
572 461210866607609
573 475592651976889
574 490348647526558
575 505486089237058
576 521012283085036
577 536934587353256
578 553260426593322
579 569997273279789
580 587152662058258
581 604734170747184
582 622749434876858
583 641206128010489
584 660111976597920
585 679474739604064
586 699302223666421
587 719602262037791
588 740382730055707
589 761651523370470
590 783416573752126
591 805685826594439
592 828467257050350
593 851768846820577
594 875598600626154
595 899964522255079
596 924874631401209
597 950336938960611
598 976359464227661
599 1002950209453776
600 1030117177412699
601 1057868345195045
602 1086211682173956
603 1115155123030642
604 1144706586113291
605 1174873945708909
606 1205665050808215
607 1237087696599984
608 1269149643677458
609 1301858588751637
610 1335222184292444
611 1369248008480895
612 1403943585301204
613 1439316353708705
614 1475373688210195
615 1512122867248405
616 1549571094266055
617 1587725465329523
618 1626592990693830
619 1666180561645293
620 1706494972609528
621 1747542887215360
622 1789330860941288
623 1831865306428571
624 1875152516686548
625 1919198629635734
626 1964009651915660
627 2009591422664545
628 2055949637927477
629 2103089813703016
630 2151017310973098
631 2199737298000372
632 2249254776027773
633 2299574540834782
634 2350701209103646
635 2402639179270463
636 2455392658582214
637 2508965623226933
638 2563361846130743
639 2618584856379097
640 2674637967752948
641 2731524237478957
642 2789246495528881
643 2847807302685929
644 2907208980659856
645 2967453569482805
646 3028542858437463
647 3090478342825046
648 3153261255732853
649 3216892524160984
650 3281372801682291
651 3346702423944313
652 3412881442219846
653 3479909578328192
654 3547786259101073
655 3616510570712255
656 3686081294113980
657 3756496858791478
658 3827755379165355
659 3899854607818599
660 3972791972891010
661 4046564530766019
662 4121169004511073
663 4196601736043217
664 4272858725610066
665 4349935583482269
666 4427827570497986
667 4506529549270567
668 4586036025851501
669 4666341100470600
670 4747438510307985
671 4829321579817147
672 4911983264628415
673 4995416101440403
674 5079612253106599
675 5164563458113973
676 5250261076840903
677 5336696040671980
678 5423858899445150
679 5511739770187560
680 5600328385803359
681 5689614043446161
682 5779585654429600
683 5870231692286645
684 5961540243918360
685 6053498957320960
686 6146095094019128
687 6239315476476153
688 6333146541790366
689 6427574288835366
690 6522584333229145
691 6618161854183309
692 6714291650789160
693 6810958088586958
694 6908145157139459
695 7005836416368480
696 7104015055421622
697 7202663838749386
698 7301765166303075
699 7401301019361722
700 7501253022027266
701 7601602386843987
702 7702329977589080
703 7803416254655222
704 7904841339170873
705 8006584958150181
706 8108626509899889
707 8210945008978195
708 8313519152881939
709 8416327266778085
710 8519347371498567
711 8622557128046053
712 8725933906847840
713 8829454732072484
714 8933096352127635
715 9036835183748460
716 9140647383762528
717 9244508792945678
718 9348395008996007
719 9452281330191897
720 9556142829555643
721 9659954298267308
722 9763690321033408
723 9867325219249981
724 9970833127512512
725 10074187936558122
726 10177363370888455
727 10280332931441733
728 10383069974340297
729 10485547653279647
730 10587738999328217
731 10689616863061737
732 10791153995382729
733 10892322989344543
734 10993096361994634
735 11093446495874170
736 11193345721807668
737 11292766260101878
738 11391680304241666
739 11490059961730183
740 11587877338692722
741 11685104480338701
742 11781713456388257
743 11877676301179640
744 11972965099875122
745 12067551928151070
746 12161408939178544
747 12254508302876326
748 12346822293578054
749 12438323228868060
750 12528983557886059
751 12618775799682379
752 12707672632154594
753 12795646829899064
754 12882671353685064
755 12968719287824486
756 13053763930133222
757 13137778728752338
758 13220737372587114
759 13302613727557896
760 13383381927422176
761 13463016309475063
762 13541491505700427
763 13618782377856223
764 13694864108945970
765 13769712137666260
766 13843302250100143
767 13915610513550433
768 13986613368402674
769 14056287561278035
770 14124610237055954
771 14191558871327486
772 14257111362481292
773 14321245963481667
774 14383941373963924
775 14445176671275633
776 14504931402574878
777 14563185515116493
778 14619919448256126
779 14675114063011110
780 14728750733921081
781 14780811277822859
782 14831278045561927
783 14880133849969280
784 14927362057331041
785 14972946514599961
786 15016871640574771
787 15059122352292240
788 15099684155916740
789 15138543072303315
790 15175685727506919
791 15211099277556439
792 15244771498541818
793 15276690710548182
794 15306845867323822
795 15335226479373283
796 15361822703122288
797 15386625263213894
798 15409625541131887
799 15430815496656959
800 15450187755959996
801 15467735532221058
802 15483452713112335
803 15497333780632956
804 15509373897951036
805 15519568828416239
806 15527915021776296
807 15534409532379142
808 15539050104692302
809 15541835090752144
810 15542763534960598
811 15541835090752144
812 15539050104692302
813 15534409532379142
814 15527915021776296
815 15519568828416239
816 15509373897951036
817 15497333780632956
818 15483452713112335
819 15467735532221058
820 15450187755959996
821 15430815496656959
822 15409625541131887
823 15386625263213894
824 15361822703122288
825 15335226479373283
826 15306845867323822
827 15276690710548182
828 15244771498541818
829 15211099277556439
830 15175685727506919
831 15138543072303315
832 15099684155916740
833 15059122352292240
834 15016871640574771
835 14972946514599961
836 14927362057331041
837 14880133849969280
838 14831278045561927
839 14780811277822859
840 14728750733921081
841 14675114063011110
842 14619919448256126
843 14563185515116493
844 14504931402574878
845 14445176671275633
846 14383941373963924
847 14321245963481667
848 14257111362481292
849 14191558871327486
850 14124610237055954
851 14056287561278035
852 13986613368402674
853 13915610513550433
854 13843302250100143
855 13769712137666260
856 13694864108945970
857 13618782377856223
858 13541491505700427
859 13463016309475063
860 13383381927422176
861 13302613727557896
862 13220737372587114
863 13137778728752338
864 13053763930133222
865 12968719287824486
866 12882671353685064
867 12795646829899064
868 12707672632154594
869 12618775799682379
870 12528983557886059
871 12438323228868060
872 12346822293578054
873 12254508302876326
874 12161408939178544
875 12067551928151070
876 11972965099875122
877 11877676301179640
878 11781713456388257
879 11685104480338701
880 11587877338692722
881 11490059961730183
882 11391680304241666
883 11292766260101878
884 11193345721807668
885 11093446495874170
886 10993096361994634
887 10892322989344543
888 10791153995382729
889 10689616863061737
890 10587738999328217
891 10485547653279647
892 10383069974340297
893 10280332931441733
894 10177363370888455
895 10074187936558122
896 9970833127512512
897 9867325219249981
898 9763690321033408
899 9659954298267308
900 9556142829555643
901 9452281330191897
902 9348395008996007
903 9244508792945678
904 9140647383762528
905 9036835183748460
906 8933096352127635
907 8829454732072484
908 8725933906847840
909 8622557128046053
910 8519347371498567
911 8416327266778085
912 8313519152881939
913 8210945008978195
914 8108626509899889
915 8006584958150181
916 7904841339170873
917 7803416254655222
918 7702329977589080
919 7601602386843987
920 7501253022027266
921 7401301019361722
922 7301765166303075
923 7202663838749386
924 7104015055421622
925 7005836416368480
926 6908145157139459
927 6810958088586958
928 6714291650789160
929 6618161854183309
930 6522584333229145
931 6427574288835366
932 6333146541790366
933 6239315476476153
934 6146095094019128
935 6053498957320960
936 5961540243918360
937 5870231692286645
938 5779585654429600
939 5689614043446161
940 5600328385803359
941 5511739770187560
942 5423858899445150
943 5336696040671980
944 5250261076840903
945 5164563458113973
946 5079612253106599
947 4995416101440403
948 4911983264628415
949 4829321579817147
950 4747438510307985
951 4666341100470600
952 4586036025851501
953 4506529549270567
954 4427827570497986
955 4349935583482269
956 4272858725610066
957 4196601736043217
958 4121169004511073
959 4046564530766019
960 3972791972891010
961 3899854607818599
962 3827755379165355
963 3756496858791478
964 3686081294113980
965 3616510570712255
966 3547786259101073
967 3479909578328192
968 3412881442219846
969 3346702423944313
970 3281372801682291
971 3216892524160984
972 3153261255732853
973 3090478342825046
974 3028542858437463
975 2967453569482805
976 2907208980659856
977 2847807302685929
978 2789246495528881
979 2731524237478957
980 2674637967752948
981 2618584856379097
982 2563361846130743
983 2508965623226933
984 2455392658582214
985 2402639179270463
986 2350701209103646
987 2299574540834782
988 2249254776027773
989 2199737298000372
990 2151017310973098
991 2103089813703016
992 2055949637927477
993 2009591422664545
994 1964009651915660
995 1919198629635734
996 1875152516686548
997 1831865306428571
998 1789330860941288
999 1747542887215360
1000 1706494972609528
1001 1666180561645293
1002 1626592990693830
1003 1587725465329523
1004 1549571094266055
1005 1512122867248405
1006 1475373688210195
1007 1439316353708705
1008 1403943585301204
1009 1369248008480895
1010 1335222184292444
1011 1301858588751637
1012 1269149643677458
1013 1237087696599984
1014 1205665050808215
1015 1174873945708909
1016 1144706586113291
1017 1115155123030642
1018 1086211682173956
1019 1057868345195045
1020 1030117177412699
1021 1002950209453776
1022 976359464227661
1023 950336938960611
1024 924874631401209
1025 899964522255079
1026 875598600626154
1027 851768846820577
1028 828467257050350
1029 805685826594439
1030 783416573752126
1031 761651523370470
1032 740382730055707
1033 719602262037791
1034 699302223666421
1035 679474739604064
1036 660111976597920
1037 641206128010489
1038 622749434876858
1039 604734170747184
1040 587152662058258
1041 569997273279789
1042 553260426593322
1043 536934587353256
1044 521012283085036
1045 505486089237058
1046 490348647526558
1047 475592651976889
1048 461210866607609
1049 447196111767398
1050 433541281180913
1051 420239328556562
1052 407283284019903
1053 394666240991089
1054 382381372002154
1055 370421915855964
1056 358781192841696
1057 347452592155562
1058 336429586543371
1059 325705719979621
1060 315274621737283
1061 305129994336046
1062 295265627053394
1063 285675384124940
1064 276353217726061
1065 267293156421037
1066 258489317615563
1067 249935896266269
1068 241627176819381
1069 233557522163930
1070 225721385084601
1071 218113297456569
1072 210727881214639
1073 203559837800120
1074 196603958660512
1075 189855114934011
1076 183308267507535
1077 176958456936318
1078 170800813062207
1079 164830545178892
1080 159042951225485
1081 153433408182971
1082 147997380868998
1083 142730412563985
1084 137628133409981
1085 132686251275763
1086 127900559773363
1087 123266929349329
1088 118781314944316
1089 114439747308027
1090 110238340304328
1091 106173282459017
1092 102240843924128
1093 98437368245632
1094 94759279010024
1095 91203071830294
1096 87765320677722
1097 84442670093739
1098 81231841219122
1099 78129624219915
1100 75132884036461
1101 72238553020681
1102 69443636407019
1103 66745205168528
1104 64140401221500
1105 61626430488492
1106 59200567856702
1107 56860150445667
1108 54602582321801
1109 52425327976532
1110 50325916807381
1111 48301936795528
1112 46351038772509
1113 44470930294083
1114 42659379693854
1115 40914210159126
1116 39233303581272
1117 37614594822327
1118 36056075379039
1119 34555787837266
1120 33111829351342
1121 31722346290416
1122 30385537541434
1123 29099649337087
1124 27862978398196
1125 26673866939713
1126 25530705653070
1127 24431928894154
1128 23376017513498
1129 22361494215565
1130 21386926250737
1131 20450920942555
1132 19552128242181
1133 18689236426307
1134 17860974520443
1135 17066108157849
1136 16303441884554
1137 15571815175325
1138 14870104620270
1139 14197220100303
1140 13552106861442
1141 12933741839899
1142 12341135635235
1143 11773328981483
1144 11229394619008
1145 10708433913262
1146 10209578630140
1147 9731987693363
1148 9274848873719
1149 8837375680898
1150 8418808965914
1151 8018413948848
1152 7635481738958
1153 7269326489361
1154 6919286843610
1155 6584723213704
1156 6265019152487
1157 5959578755669
1158 5667827963566
1159 5389212078927
1160 5123197006289
1161 4869266881582
1162 4626925247931
1163 4395692797816
1164 4175108488550
1165 3964727389027
1166 3764121741811
1167 3572878910940
1168 3390602389686
1169 3216909849508
1170 3051434096031
1171 2893821211946
1172 2743731467669
1173 2600837554495
1174 2464825448513
1175 2335392734382
1176 2212249424868
1177 2095116368246
1178 1983726028916
1179 1877820975168
1180 1777154619819
1181 1681489788194
1182 1590599420536
1183 1504265214046
1184 1422278292091
1185 1344437917067
1186 1270552124993
1187 1200436509276
1188 1133914822434
1189 1070817824773
1190 1010983857544
1191 954257753733
1192 900491381529
1193 849542617012
1194 801275859118
1195 755561059108
1196 712274211009
1197 671296435063
1198 632514442353
1199 595819672189
1200 561108732221
1201 528282584833
1202 497246966079
1203 467911618783
1204 440190689221
1205 414002006749
1206 389267459099
1207 365912314318
1208 343865577844
1209 323059354425
1210 303429185857
1211 284913452879
1212 267453694512
1213 250994045973
1214 235481542190
1215 220865589927
1216 207098254570
1217 194133766301
1218 181928790813
1219 170441966142
1220 159634159836
1221 149468034620
1222 139908291043
1223 130921262045
1224 122475141756
1225 114539605867
1226 107086028628
1227 100087127594
1228 93517168095
1229 87351632249
1230 81567411419
1231 76142496873
1232 71056162189
1233 66288674187
1234 61821464449
1235 57636860670
1236 53718247833
1237 50049817550
1238 46616720518
1239 43404832757
1240 40400898786
1241 37592314751
1242 34967262684
1243 32514508546
1244 30223529082
1245 28084323773
1246 26087533643
1247 24224267278
1248 22486212019
1249 20865472184
1250 19354673930
1251 17946814901
1252 16635362300
1253 15414114032
1254 14277290236
1255 13219404469
1256 12235349889
1257 11320279729
1258 10469687646
1259 9679297643
1260 8945138936
1261 8263443950
1262 7630718671
1263 7043648186
1264 6499162145
1265 5994347965
1266 5526511602
1267 5093097335
1268 4691744778
1269 4320214692
1270 3976441876
1271 3658467191
1272 3364486581
1273 3092788314
1274 2841798846
1275 2610024936
1276 2396096082
1277 2198711618
1278 2016679881
1279 1848869496
1280 1694245990
1281 1551826889
1282 1420715489
1283 1300059954
1284 1189084411
1285 1087051347
1286 993290571
1287 907164667
1288 828095629
1289 755533468
1290 688980643
1291 627963266
1292 572053851
1293 520844871
1294 473969587
1295 431078138
1296 391856581
1297 356005000
1298 323255164
1299 293350487
1300 266062168
1301 241171101
1302 218482569
1303 197809729
1304 178987225
1305 161856093
1306 146276124
1307 132112328
1308 119246146
1309 107563103
1310 96963169
1311 87349501
1312 78637842
1313 70746444
1314 63604540
1315 57143201
1316 51303148
1317 46026431
1318 41263462
1319 36965620
1320 33091578
1321 29600587
1322 26458297
1323 23630664
1324 21089176
1325 18805464
1326 16755957
1327 14917024
1328 13269250
1329 11793028
1330 10472375
1331 9291060
1332 8236001
1333 7293767
1334 6453684
1335 5704686
1336 5038070
1337 4444748
1338 3917670
1339 3449359
1340 3034135
1341 2665885
1342 2340024
1343 2051569
1344 1796855
1345 1571812
1346 1373524
1347 1198689
1348 1044984
1349 909741
1350 791131
1351 686983
1352 595872
1353 516054
1354 446405
1355 385528
1356 332557
1357 286364
1358 246288
1359 211428
1360 181274
1361 155112
1362 132559
1363 113039
1364 96271
1365 81801
1366 69414
1367 58754
1368 49668
1369 41869
1370 35251
1371 29588
1372 24803
1373 20722
1374 17293
1375 14375
1376 11937
1377 9871
1378 8154
1379 6703
1380 5507
1381 4498
1382 3673
1383 2980
1384 2417
1385 1946
1386 1568
1387 1251
1388 1000
1389 791
1390 627
1391 490
1392 385
1393 297
1394 231
1395 176
1396 135
1397 101
1398 77
1399 56
1400 42
1401 30
1402 22
1403 15
1404 11
1405 7
1406 5
1407 3
1408 2
1409 1
1410 1
Sum: 3535316142212174320

Comments:

There is a direct correspondence between this method and the method others have posted of expanding a very large polynomial and looking at certain coefficients. But I wrote this code before realizing how direct the correspondence was, and also no one else has really explained how an efficient calculation of the coefficients works, so here we are.

With regard to the part of the comment by ThatDonGuy specifically about looking at all 280 products, this is not necessary because the largest term, expanding the full polynomial that includes sums of more the 20 balls, is x80y1410. If you collect like terms, there are "only" 81*1411=114291 terms in the full product. If you collect like terms at each stage (each multiplication by (1+xyn)), the number of terms remains manageable at each stage.

However, I think ThatDonGuy has a great explanation of why the polynomial expansion works and I have nothing to add to it. (I just have an explanation of how to efficiently do this calculation.)
weezrDASvegas
weezrDASvegas
Joined: Feb 2, 2018
  • Threads: 2
  • Posts: 69
May 21st, 2019 at 2:25:23 AM permalink
Quote: Wizard

I am doing a software review of a company that offers prop bets on the sum of the balls in keno. As a reminder, in keno the game draws 20 numbers from a 80 balls numbered 1 to 80. The problem at hand is finding a probability of any given total without using a simulation.

I do know simulations are a perfectly valid way of analyzing casino games, but I also feel more satisfied finding an exact answer. At first I wrote a simple program to loop through 3,535,316,142,212,180,000 possible ways to choose 20 out of 80 balls. Needless to say, it would have taken centuries to cycle through them all. I've also toyed with shortcut code, but I fear I'll die before the program is done running.

I've also just counted combinations by hand, hoping to find a pattern (maybe some variant of Fibonaci), but so far have not found anything. So, at this point I open it up to the forum. You do not have to put replies in spoiler tags and may refer to any outside sources. This isn't the usual math puzzle thread.

The question for the poll is what are your thoughts on it?

In this spoiler tag are the results of my simulation. I excluded extreme totals where the probability was less than 1 in a million. It's a big table, so I won't clutter up the thread with it, unless you want to see it. The third column is the expected count per million games.




Aint a formula for that. gotta do it in soft code - iterate thru all combos in the game.

There is a website that shows the sums for many lottery games but no keno. probably keno has too many combos compared 2 lottos, even mega mills or powerball.

Total powerball combos (5 of 69 & 1 26)=292,201,338
Total keno combos (20 from 80)=3.53531614221217E+18 (19 digit number... very, very large for the computers).

But looks like the right algo exists and applicable to any lotto game.

https://saliu.com/forum/lottery-sums.html

Powerball 5/69 & 1/26 Lottery SUMS Chart



Sum-total Combinations Percentage

16 1 0.00%
17 2 0.00%
18 4 0.00%
19 7 0.00%
20 12 0.00%
21 19 0.00%
22 29 0.00%
23 42 0.00%
24 60 0.00%
25 83 0.00%
26 113 0.00%
27 150 0.00%
28 197 0.00%
29 254 0.00%
30 324 0.00%
31 408 0.00%
32 509 0.00%
33 628 0.00%
34 769 0.00%
35 933 0.00%
36 1125 0.00%
37 1346 0.00%
38 1601 0.00%
39 1892 0.00%
40 2225 0.00%
41 2602 0.00%
42 3028 0.00%
43 3507 0.00%
44 4045 0.00%
45 4645 0.00%
46 5314 0.00%
47 6055 0.00%
48 6876 0.00%
49 7781 0.00%
50 8777 0.00%
51 9869 0.00%
52 11065 0.00%
53 12370 0.00%
54 13792 0.00%
55 15337 0.01%
56 17014 0.01%
57 18828 0.01%
58 20789 0.01%
59 22903 0.01%
60 25180 0.01%
61 27627 0.01%
62 30253 0.01%
63 33066 0.01%
64 36077 0.01%
65 39293 0.01%
66 42725 0.01%
67 46381 0.02%
68 50273 0.02%
69 54409 0.02%
70 58801 0.02%
71 63458 0.02%
72 68392 0.02%
73 73613 0.03%
74 79133 0.03%
75 84962 0.03%
76 91114 0.03%
77 97598 0.03%
78 104428 0.04%
79 111615 0.04%
80 119173 0.04%
81 127112 0.04%
82 135446 0.05%
83 144186 0.05%
84 153346 0.05%
85 162936 0.06%
86 172970 0.06%
87 183457 0.06%
88 194412 0.07%
89 205843 0.07%
90 217763 0.07%
91 230181 0.08%
92 243109 0.08%
93 256554 0.09%
94 270528 0.09%
95 285037 0.10%
96 300092 0.10%
97 315697 0.11%
98 331862 0.11%
99 348590 0.12%
100 365890 0.13%
101 383763 0.13%
102 402215 0.14%
103 421247 0.14%
104 440864 0.15%
105 461063 0.16%
106 481848 0.16%
107 503215 0.17%
108 525166 0.18%
109 547695 0.19%
110 570802 0.20%
111 594480 0.20%
112 618727 0.21%
113 643534 0.22%
114 668897 0.23%
115 694806 0.24%
116 721256 0.25%
117 748233 0.26%
118 775731 0.27%
119 803736 0.28%
120 832239 0.28%
121 861224 0.29%
122 890680 0.30%
123 920590 0.32%
124 950942 0.33%
125 981716 0.34%
126 1012898 0.35%
127 1044467 0.36%
128 1076408 0.37%
129 1108697 0.38%
130 1141317 0.39%
131 1174244 0.40%
132 1207458 0.41%
133 1240933 0.42%
134 1274648 0.44%
135 1308575 0.45%
136 1342692 0.46%
137 1376968 0.47%
138 1411379 0.48%
139 1445894 0.49%
140 1480487 0.51%
141 1515124 0.52%
142 1549777 0.53%
143 1584412 0.54%
144 1618999 0.55%
145 1653501 0.57%
146 1687887 0.58%
147 1722119 0.59%
148 1756165 0.60%
149 1789986 0.61%
150 1823548 0.62%
151 1856813 0.64%
152 1889746 0.65%
153 1922308 0.66%
154 1954464 0.67%
155 1986176 0.68%
156 2017409 0.69%
157 2048124 0.70%
158 2078287 0.71%
159 2107861 0.72%
160 2136812 0.73%
161 2165104 0.74%
162 2192703 0.75%
163 2219576 0.76%
164 2245691 0.77%
165 2271015 0.78%
166 2295518 0.79%
167 2319170 0.79%
168 2341943 0.80%
169 2363809 0.81%
170 2384742 0.82%
171 2404718 0.82%
172 2423713 0.83%
173 2441705 0.84%
174 2458673 0.84%
175 2474598 0.85%
176 2489462 0.85%
177 2503247 0.86%
178 2515938 0.86%
179 2527521 0.86%
180 2537983 0.87%
181 2547312 0.87%
182 2555497 0.87%
183 2562530 0.88%
184 2568403 0.88%
185 2573109 0.88%
186 2576643 0.88%
187 2579001 0.88%

188 2580181 0.88%
189 2580181 0.88%

190 2579001 0.88%
191 2576643 0.88%
192 2573109 0.88%
193 2568403 0.88%
194 2562530 0.88%
195 2555497 0.87%
196 2547312 0.87%
197 2537983 0.87%
198 2527521 0.86%
199 2515938 0.86%
200 2503247 0.86%
201 2489462 0.85%
202 2474598 0.85%
203 2458673 0.84%
204 2441705 0.84%
205 2423713 0.83%
206 2404718 0.82%
207 2384742 0.82%
208 2363809 0.81%
209 2341943 0.80%
210 2319170 0.79%
211 2295518 0.79%
212 2271015 0.78%
213 2245691 0.77%
214 2219576 0.76%
215 2192703 0.75%
216 2165104 0.74%
217 2136812 0.73%
218 2107861 0.72%
219 2078287 0.71%
220 2048124 0.70%
221 2017409 0.69%
222 1986176 0.68%
223 1954464 0.67%
224 1922308 0.66%
225 1889746 0.65%
226 1856813 0.64%
227 1823548 0.62%
228 1789986 0.61%
229 1756165 0.60%
230 1722119 0.59%
231 1687887 0.58%
232 1653501 0.57%
233 1618999 0.55%
234 1584412 0.54%
235 1549777 0.53%
236 1515124 0.52%
237 1480487 0.51%
238 1445894 0.49%
239 1411379 0.48%
240 1376968 0.47%
241 1342692 0.46%
242 1308575 0.45%
243 1274648 0.44%
244 1240933 0.42%
245 1207458 0.41%
246 1174244 0.40%
247 1141317 0.39%
248 1108697 0.38%
249 1076408 0.37%
250 1044467 0.36%
251 1012898 0.35%
252 981716 0.34%
253 950942 0.33%
254 920590 0.32%
255 890680 0.30%
256 861224 0.29%
257 832239 0.28%
258 803736 0.28%
259 775731 0.27%
260 748233 0.26%
261 721256 0.25%
262 694806 0.24%
263 668897 0.23%
264 643534 0.22%
265 618727 0.21%
266 594480 0.20%
267 570802 0.20%
268 547695 0.19%
269 525166 0.18%
270 503215 0.17%
271 481848 0.16%
272 461063 0.16%
273 440864 0.15%
274 421247 0.14%
275 402215 0.14%
276 383763 0.13%
277 365890 0.13%
278 348590 0.12%
279 331862 0.11%
280 315697 0.11%
281 300092 0.10%
282 285037 0.10%
283 270528 0.09%
284 256554 0.09%
285 243109 0.08%
286 230181 0.08%
287 217763 0.07%
288 205843 0.07%
289 194412 0.07%
290 183457 0.06%
291 172970 0.06%
292 162936 0.06%
293 153346 0.05%
294 144186 0.05%
295 135446 0.05%
296 127112 0.04%
297 119173 0.04%
298 111615 0.04%
299 104428 0.04%
300 97598 0.03%
301 91114 0.03%
302 84962 0.03%
303 79133 0.03%
304 73613 0.03%
305 68392 0.02%
306 63458 0.02%
307 58801 0.02%
308 54409 0.02%
309 50273 0.02%
310 46381 0.02%
311 42725 0.01%
312 39293 0.01%
313 36077 0.01%
314 33066 0.01%
315 30253 0.01%
316 27627 0.01%
317 25180 0.01%
318 22903 0.01%
319 20789 0.01%
320 18828 0.01%
321 17014 0.01%
322 15337 0.01%
323 13792 0.00%
324 12370 0.00%
325 11065 0.00%
326 9869 0.00%
327 8777 0.00%
328 7781 0.00%
329 6876 0.00%
330 6055 0.00%
331 5314 0.00%
332 4645 0.00%
333 4045 0.00%
334 3507 0.00%
335 3028 0.00%
336 2602 0.00%
337 2225 0.00%
338 1892 0.00%
339 1601 0.00%
340 1346 0.00%
341 1125 0.00%
342 933 0.00%
343 769 0.00%
344 628 0.00%
345 509 0.00%
346 408 0.00%
347 324 0.00%
348 254 0.00%
349 197 0.00%
350 150 0.00%
351 113 0.00%
352 83 0.00%
353 60 0.00%
354 42 0.00%
355 29 0.00%
356 19 0.00%
357 12 0.00%
358 7 0.00%
359 4 0.00%
360 2 0.00%
361 1 0.00%

Total: 346 292201338 100%


Quote: Wizard

Note: Post formatting corrected by management.

Last edited by: unnamed administrator on May 21, 2019
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1324
  • Posts: 21759
May 21st, 2019 at 6:40:07 AM permalink
I get the theory about finding the generating function, but to actually do it would be the equal amount of work as just cycling through all combin(80,20) balls by brute force.

ZPP, thanks for the code, I think I will try that method next. I started to go down that road, but didn't finishing, thinking it would only cut down centuries of computer time to years. Let's see what happens....
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
May 21st, 2019 at 7:28:40 AM permalink
Quote: Wizard

I get the theory about finding the generating function, but to actually do it would be the equal amount of work as just cycling through all combin(80,20) balls by brute force.

the generating function is also easily changed for the values on the balls. say instead of 1 to 80 they were 0 to 79. simple adjustment

it takes 0ms on my 10 year old Dell 2 core laptop for all the calculations (according to pari/gp calculator) and it is actually doing all the draws from 0 to 80 inclusive and prints to a text file.

from there it is easy (for me) to get what is needed into Excel or work on the results further in pari.
generating functions go way back and it makes sense why.(they had no computers to do it then)
even poker gurus now (like BA) uses GFs to describe getting particular hands and so on.

of course there are 'expert computer guys and gals' that can do excellent code for almost any problem.

some get (and can use) generating functions and add them to their toolbox, others never get them for whatever reason(s).

added: reminds me of learning inclusion-exclusion and wondering why we have to do it this way. add and subtract over-counts seemed way confusing and not required as they were always another way to solve the problem. Most now, that even do not understand PIE, still use it in a computer to solve some problems because it IS so easy and follows basic counting methods. example code for coupon collecting problem: n=6;\\number of coupons
t=10;\\number of draws
x=sum(k=0,n,(-1)^k*binomial(n,k)*(1-(k/n))^t);

this was so confusing at 1st: (-1)^k*binomial(n,k)*(1-(k/n))^t
Last edited by: 7craps on May 21, 2019
winsome johnny (not Win some johnny)
charliepatrick
charliepatrick
Joined: Jun 17, 2011
  • Threads: 31
  • Posts: 2049
Thanks for this post from:
RS
May 21st, 2019 at 7:34:51 AM permalink
I'm about to go out (it's afternoon shoping here) but here's an idea I might look at later.
(i) Split the numbers into blocks 1-10, 11-20 etc. (consider 1-10 as 0*10+1 thru 0*10+10, 2-20 at 1*10+1 thru 1*10+10...)
(ii) Similar to working out poker hands, you can have a pattern of (a) 10 in one block, 10 in another (b) 10 9 1 (c) 10 8 2 (d) 10 8 1 1 etc.
(iii) Work out the permutations for various numbers of balls within a block for the digits. (e..g 1 number in block can be 1:1 2:1 ... 10:1).
(iv) Work out the permutations for various blocks being used, this will only affect the tens value of the total.
(v) Add together the digits part from (iii) and the tens part from (iv).
e.g. Looking at {10,10}, the sum of the digits part is fixed at 55+55=110. The sum of the tens part can be 10*0+10*10, 10*0+10*20, 10*10+10*10 etc. This gets 1 combination for 210 etc.
e.g. Looking at (10,9,1), the sum of the digits part can be {55}+{45,47,48,49...54)+(1,2,3,4,...,9,10}. Then the tens can be 10*0+9*10+1*20 etc.
gordonm888
gordonm888
Joined: Feb 18, 2015
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May 21st, 2019 at 10:21:34 AM permalink
Quote: charliepatrick

I'm about to go out (it's afternoon shoping here) but here's an idea I might look at later.
(i) Split the numbers into blocks 1-10, 11-20 etc. (consider 1-10 as 0*10+1 thru 0*10+10, 2-20 at 1*10+1 thru 1*10+10...)
(ii) Similar to working out poker hands, you can have a pattern of (a) 10 in one block, 10 in another (b) 10 9 1 (c) 10 8 2 (d) 10 8 1 1 etc.
(iii) Work out the permutations for various numbers of balls within a block for the digits. (e..g 1 number in block can be 1:1 2:1 ... 10:1).
(iv) Work out the permutations for various blocks being used, this will only affect the tens value of the total.
(v) Add together the digits part from (iii) and the tens part from (iv).
e.g. Looking at {10,10}, the sum of the digits part is fixed at 55+55=110. The sum of the tens part can be 10*0+10*10, 10*0+10*20, 10*10+10*10 etc. This gets 1 combination for 210 etc.
e.g. Looking at (10,9,1), the sum of the digits part can be {55}+{45,47,48,49...54)+(1,2,3,4,...,9,10}. Then the tens can be 10*0+9*10+1*20 etc.



I like this approach, ingenious. I wonder whether it might be slightly more efficient in base 9.

edit: Or 4-tier it in base 3.
Last edited by: gordonm888 on May 21, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.

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