# Keno Sums

## Introduction

This page shows the probability of any given sum of the number of 20 keno balls. For those not familiar with keno, it is a game in which the player bets on the draw of 20 balls, without replacement, numbered 1 to 80. I created this page when I noticed bets on the sum of the draw in Blitz Keno.

There are combin(80,20)= 3,535,316,142,212,174,320 ways to choose 20 items out of 80. The following table shows the number of combinations for any given possible total from 210 to 1,410, the probability of that total, and the probability of that total or less.

### Keno Sum of Ball Draw

Total Combinations Probability
(Exact)
Probability
(This Total or Less)
210 1 0.000000000000 0.000000000000
211 1 0.000000000000 0.000000000000
212 2 0.000000000000 0.000000000000
213 3 0.000000000000 0.000000000000
214 5 0.000000000000 0.000000000000
215 7 0.000000000000 0.000000000000
216 11 0.000000000000 0.000000000000
217 15 0.000000000000 0.000000000000
218 22 0.000000000000 0.000000000000
219 30 0.000000000000 0.000000000000
220 42 0.000000000000 0.000000000000
221 56 0.000000000000 0.000000000000
222 77 0.000000000000 0.000000000000
223 101 0.000000000000 0.000000000000
224 135 0.000000000000 0.000000000000
225 176 0.000000000000 0.000000000000
226 231 0.000000000000 0.000000000000
227 297 0.000000000000 0.000000000000
228 385 0.000000000000 0.000000000000
229 490 0.000000000000 0.000000000000
230 627 0.000000000000 0.000000000000
231 791 0.000000000000 0.000000000000
232 1000 0.000000000000 0.000000000000
233 1251 0.000000000000 0.000000000000
234 1568 0.000000000000 0.000000000000
235 1946 0.000000000000 0.000000000000
236 2417 0.000000000000 0.000000000000
237 2980 0.000000000000 0.000000000000
238 3673 0.000000000000 0.000000000000
239 4498 0.000000000000 0.000000000000
240 5507 0.000000000000 0.000000000000
241 6703 0.000000000000 0.000000000000
242 8154 0.000000000000 0.000000000000
243 9871 0.000000000000 0.000000000000
244 11937 0.000000000000 0.000000000000
245 14375 0.000000000000 0.000000000000
246 17293 0.000000000000 0.000000000000
247 20722 0.000000000000 0.000000000000
248 24803 0.000000000000 0.000000000000
249 29588 0.000000000000 0.000000000000
250 35251 0.000000000000 0.000000000000
251 41869 0.000000000000 0.000000000000
252 49668 0.000000000000 0.000000000000
253 58754 0.000000000000 0.000000000000
254 69414 0.000000000000 0.000000000000
255 81801 0.000000000000 0.000000000000
256 96271 0.000000000000 0.000000000000
257 113039 0.000000000000 0.000000000000
258 132559 0.000000000000 0.000000000000
259 155112 0.000000000000 0.000000000000
260 181274 0.000000000000 0.000000000000
261 211428 0.000000000000 0.000000000000
262 246288 0.000000000000 0.000000000000
263 286364 0.000000000000 0.000000000001
264 332557 0.000000000000 0.000000000001
265 385528 0.000000000000 0.000000000001
266 446405 0.000000000000 0.000000000001
267 516054 0.000000000000 0.000000000001
268 595872 0.000000000000 0.000000000001
269 686983 0.000000000000 0.000000000001
270 791131 0.000000000000 0.000000000002
271 909741 0.000000000000 0.000000000002
272 1044984 0.000000000000 0.000000000002
273 1198689 0.000000000000 0.000000000003
274 1373524 0.000000000000 0.000000000003
275 1571812 0.000000000000 0.000000000003
276 1796855 0.000000000001 0.000000000004
277 2051569 0.000000000001 0.000000000004
278 2340024 0.000000000001 0.000000000005
279 2665885 0.000000000001 0.000000000006
280 3034135 0.000000000001 0.000000000007
281 3449359 0.000000000001 0.000000000008
282 3917670 0.000000000001 0.000000000009
283 4444748 0.000000000001 0.000000000010
284 5038070 0.000000000001 0.000000000011
285 5704686 0.000000000002 0.000000000013
286 6453684 0.000000000002 0.000000000015
287 7293767 0.000000000002 0.000000000017
288 8236001 0.000000000002 0.000000000019
289 9291060 0.000000000003 0.000000000022
290 10472375 0.000000000003 0.000000000025
291 11793028 0.000000000003 0.000000000028
292 13269250 0.000000000004 0.000000000032
293 14917024 0.000000000004 0.000000000036
294 16755957 0.000000000005 0.000000000041
295 18805464 0.000000000005 0.000000000046
296 21089176 0.000000000006 0.000000000052
297 23630664 0.000000000007 0.000000000059
298 26458297 0.000000000007 0.000000000066
299 29600587 0.000000000008 0.000000000075
300 33091578 0.000000000009 0.000000000084
301 36965620 0.000000000010 0.000000000095
302 41263462 0.000000000012 0.000000000106
303 46026431 0.000000000013 0.000000000119
304 51303148 0.000000000015 0.000000000134
305 57143201 0.000000000016 0.000000000150
306 63604540 0.000000000018 0.000000000168
307 70746444 0.000000000020 0.000000000188
308 78637842 0.000000000022 0.000000000210
309 87349501 0.000000000025 0.000000000235
310 96963169 0.000000000027 0.000000000262
311 107563103 0.000000000030 0.000000000293
312 119246146 0.000000000034 0.000000000326
313 132112328 0.000000000037 0.000000000364
314 146276124 0.000000000041 0.000000000405
315 161856093 0.000000000046 0.000000000451
316 178987225 0.000000000051 0.000000000502
317 197809729 0.000000000056 0.000000000558
318 218482569 0.000000000062 0.000000000619
319 241171101 0.000000000068 0.000000000688
320 266062168 0.000000000075 0.000000000763
321 293350487 0.000000000083 0.000000000846
322 323255164 0.000000000091 0.000000000937
323 356005000 0.000000000101 0.000000001038
324 391856581 0.000000000111 0.000000001149
325 431078138 0.000000000122 0.000000001271
326 473969587 0.000000000134 0.000000001405
327 520844871 0.000000000147 0.000000001552
328 572053851 0.000000000162 0.000000001714
329 627963266 0.000000000178 0.000000001892
330 688980643 0.000000000195 0.000000002086
331 755533468 0.000000000214 0.000000002300
332 828095629 0.000000000234 0.000000002534
333 907164667 0.000000000257 0.000000002791
334 993290571 0.000000000281 0.000000003072
335 1087051347 0.000000000307 0.000000003379
336 1189084411 0.000000000336 0.000000003716
337 1300059954 0.000000000368 0.000000004084
338 1420715489 0.000000000402 0.000000004485
339 1551826889 0.000000000439 0.000000004924
340 1694245990 0.000000000479 0.000000005404
341 1848869496 0.000000000523 0.000000005927
342 2016679881 0.000000000570 0.000000006497
343 2198711618 0.000000000622 0.000000007119
344 2396096082 0.000000000678 0.000000007797
345 2610024936 0.000000000738 0.000000008535
346 2841798846 0.000000000804 0.000000009339
347 3092788314 0.000000000875 0.000000010214
348 3364486581 0.000000000952 0.000000011165
349 3658467191 0.000000001035 0.000000012200
350 3976441876 0.000000001125 0.000000013325
351 4320214692 0.000000001222 0.000000014547
352 4691744778 0.000000001327 0.000000015874
353 5093097335 0.000000001441 0.000000017315
354 5526511602 0.000000001563 0.000000018878
355 5994347965 0.000000001696 0.000000020573
356 6499162145 0.000000001838 0.000000022412
357 7043648186 0.000000001992 0.000000024404
358 7630718671 0.000000002158 0.000000026563
359 8263443950 0.000000002337 0.000000028900
360 8945138936 0.000000002530 0.000000031430
361 9679297643 0.000000002738 0.000000034168
362 10469687646 0.000000002961 0.000000037130
363 11320279729 0.000000003202 0.000000040332
364 12235349889 0.000000003461 0.000000043793
365 13219404469 0.000000003739 0.000000047532
366 14277290236 0.000000004038 0.000000051570
367 15414114032 0.000000004360 0.000000055930
368 16635362300 0.000000004705 0.000000060636
369 17946814901 0.000000005076 0.000000065712
370 19354673930 0.000000005475 0.000000071187
371 20865472184 0.000000005902 0.000000077089
372 22486212019 0.000000006360 0.000000083449
373 24224267278 0.000000006852 0.000000090301
374 26087533643 0.000000007379 0.000000097681
375 28084323773 0.000000007944 0.000000105624
376 30223529082 0.000000008549 0.000000114173
377 32514508546 0.000000009197 0.000000123371
378 34967262684 0.000000009891 0.000000133261
379 37592314751 0.000000010633 0.000000143895
380 40400898786 0.000000011428 0.000000155323
381 43404832757 0.000000012277 0.000000167600
382 46616720518 0.000000013186 0.000000180786
383 50049817550 0.000000014157 0.000000194943
384 53718247833 0.000000015195 0.000000210138
385 57636860670 0.000000016303 0.000000226441
386 61821464449 0.000000017487 0.000000243928
387 66288674187 0.000000018750 0.000000262678
388 71056162189 0.000000020099 0.000000282777
389 76142496873 0.000000021538 0.000000304315
390 81567411419 0.000000023072 0.000000327387
391 87351632249 0.000000024708 0.000000352095
392 93517168095 0.000000026452 0.000000378548
393 100087127594 0.000000028311 0.000000406858
394 107086028628 0.000000030290 0.000000437149
395 114539605867 0.000000032399 0.000000469547
396 122475141756 0.000000034643 0.000000504191
397 130921262045 0.000000037032 0.000000541223
398 139908291043 0.000000039574 0.000000580798
399 149468034620 0.000000042279 0.000000623076
400 159634159836 0.000000045154 0.000000668230
401 170441966142 0.000000048211 0.000000716442
402 181928790813 0.000000051460 0.000000767902
403 194133766301 0.000000054913 0.000000822815
404 207098254570 0.000000058580 0.000000881395
405 220865589927 0.000000062474 0.000000943869
406 235481542190 0.000000066608 0.000001010477
407 250994045973 0.000000070996 0.000001081473
408 267453694512 0.000000075652 0.000001157125
409 284913452879 0.000000080591 0.000001237716
410 303429185857 0.000000085828 0.000001323544
411 323059354425 0.000000091381 0.000001414924
412 343865577844 0.000000097266 0.000001512190
413 365912314318 0.000000103502 0.000001615692
414 389267459099 0.000000110108 0.000001725801
415 414002006749 0.000000117105 0.000001842905
416 440190689221 0.000000124512 0.000001967418
417 467911618783 0.000000132354 0.000002099771
418 497246966079 0.000000140651 0.000002240422
419 528282584833 0.000000149430 0.000002389853
420 561108732221 0.000000158715 0.000002548568
421 595819672189 0.000000168534 0.000002717101
422 632514442353 0.000000178913 0.000002896015
423 671296435063 0.000000189883 0.000003085898
424 712274211009 0.000000201474 0.000003287372
425 755561059108 0.000000213718 0.000003501090
426 801275859118 0.000000226649 0.000003727739
427 849542617012 0.000000240302 0.000003968040
428 900491381529 0.000000254713 0.000004222754
429 954257753733 0.000000269921 0.000004492675
430 1010983857544 0.000000285967 0.000004778642
431 1070817824773 0.000000302892 0.000005081534
432 1133914822434 0.000000320739 0.000005402273
433 1200436509276 0.000000339556 0.000005741829
434 1270552124993 0.000000359389 0.000006101217
435 1344437917067 0.000000380288 0.000006481505
436 1422278292091 0.000000402306 0.000006883811
437 1504265214046 0.000000425497 0.000007309308
438 1590599420536 0.000000449917 0.000007759225
439 1681489788194 0.000000475626 0.000008234851
440 1777154619819 0.000000502686 0.000008737537
441 1877820975168 0.000000531161 0.000009268698
442 1983726028916 0.000000561117 0.000009829815
443 2095116368246 0.000000592625 0.000010422440
444 2212249424868 0.000000625757 0.000011048197
445 2335392734382 0.000000660590 0.000011708787
446 2464825448513 0.000000697201 0.000012405988
447 2600837554495 0.000000735673 0.000013141661
448 2743731467669 0.000000776092 0.000013917753
449 2893821211946 0.000000818547 0.000014736300
450 3051434096031 0.000000863129 0.000015599429
451 3216909849508 0.000000909936 0.000016509364
452 3390602389686 0.000000959066 0.000017468431
453 3572878910940 0.000001010625 0.000018479056
454 3764121741811 0.000001064720 0.000019543776
455 3964727389027 0.000001121463 0.000020665239
456 4175108488550 0.000001180972 0.000021846211
457 4395692797816 0.000001243366 0.000023089577
458 4626925247931 0.000001308773 0.000024398350
459 4869266881582 0.000001377321 0.000025775671
460 5123197006289 0.000001449148 0.000027224819
461 5389212078927 0.000001524393 0.000028749212
462 5667827963566 0.000001603203 0.000030352415
463 5959578755669 0.000001685727 0.000032038142
464 6265019152487 0.000001772124 0.000033810266
465 6584723213704 0.000001862556 0.000035672822
466 6919286843610 0.000001957190 0.000037630013
467 7269326489361 0.000002056203 0.000039686215
468 7635481738958 0.000002159773 0.000041845989
469 8018413948848 0.000002268090 0.000044114078
470 8418808965914 0.000002381345 0.000046495424
471 8837375680898 0.000002499741 0.000048995165
472 9274848873719 0.000002623485 0.000051618650
473 9731987693363 0.000002752791 0.000054371441
474 10209578630140 0.000002887883 0.000057259324
475 10708433913262 0.000003028989 0.000060288313
476 11229394619008 0.000003176348 0.000063464661
477 11773328981483 0.000003330205 0.000066794867
478 12341135635235 0.000003490815 0.000070285682
479 12933741839899 0.000003658440 0.000073944122
480 13552106861442 0.000003833351 0.000077777473
481 14197220100303 0.000004015828 0.000081793300
482 14870104620270 0.000004206160 0.000085999460
483 15571815175325 0.000004404646 0.000090404106
484 16303441884554 0.000004611594 0.000095015700
485 17066108157849 0.000004827322 0.000099843021
486 17860974520443 0.000005052158 0.000104895179
487 18689236426307 0.000005286440 0.000110181619
488 19552128242181 0.000005530518 0.000115712136
489 20450920942555 0.000005784750 0.000121496887
490 21386926250737 0.000006049509 0.000127546396
491 22361494215565 0.000006325175 0.000133871571
492 23376017513498 0.000006612143 0.000140483714
493 24431928894154 0.000006910819 0.000147394533
494 25530705653070 0.000007221619 0.000154616152
495 26673866939713 0.000007544974 0.000162161125
496 27862978398196 0.000007881326 0.000170042451
497 29099649337087 0.000008231131 0.000178273582
498 30385537541434 0.000008594857 0.000186868439
499 31722346290416 0.000008972987 0.000195841426
500 33111829351342 0.000009366017 0.000205207443
501 34555787837266 0.000009774455 0.000214981898
502 36056075379039 0.000010198826 0.000225180724
503 37614594822327 0.000010639669 0.000235820393
504 39233303581272 0.000011097538 0.000246917931
505 40914210159126 0.000011572999 0.000258490930
506 42659379693854 0.000012066638 0.000270557568
507 44470930294083 0.000012579053 0.000283136621
508 46351038772509 0.000013110861 0.000296247482
509 48301936795528 0.000013662692 0.000309910175
510 50325916807381 0.000014235196 0.000324145370
511 52425327976532 0.000014829035 0.000338974405
512 54602582321801 0.000015444894 0.000354419299
513 56860150445667 0.000016083470 0.000370502769
514 59200567856702 0.000016745481 0.000387248250
515 61626430488492 0.000017431660 0.000404679910
516 64140401221500 0.000018142763 0.000422822673
517 66745205168528 0.000018879558 0.000441702230
518 69443636407019 0.000019642836 0.000461345067
519 72238553020681 0.000020433407 0.000481778473
520 75132884036461 0.000021252098 0.000503030571
521 78129624219915 0.000022099756 0.000525130327
522 81231841219122 0.000022977250 0.000548107577
523 84442670093739 0.000023885465 0.000571993042
524 87765320677722 0.000024825310 0.000596818352
525 91203071830294 0.000025797713 0.000622616065
526 94759279010024 0.000026803622 0.000649419688
527 98437368245632 0.000027844007 0.000677263695
528 102240843924128 0.000028919859 0.000706183554
529 106173282459017 0.000030032189 0.000736215743
530 110238340304328 0.000031182032 0.000767397775
531 114439747308027 0.000032370442 0.000799768217
532 118781314944316 0.000033598499 0.000833366716
533 123266929349329 0.000034867300 0.000868234016
534 127900559773363 0.000036177970 0.000904411986
535 132686251275763 0.000037531651 0.000941943636
536 137628133409981 0.000038929512 0.000980873149
537 142730412563985 0.000040372744 0.001021245893
538 147997380868998 0.000041862559 0.001063108452
539 153433408182971 0.000043400194 0.001106508646
540 159042951225485 0.000044986911 0.001151495557
541 164830545178892 0.000046623990 0.001198119547
542 170800813062207 0.000048312741 0.001246432288
543 176958456936318 0.000050054493 0.001296486781
544 183308267507535 0.000051850601 0.001348337382
545 189855114934011 0.000053702443 0.001402039825
546 196603958660512 0.000055611422 0.001457651248
547 203559837800120 0.000057578963 0.001515230211
548 210727881214639 0.000059606517 0.001574836728
549 218113297456569 0.000061695557 0.001636532284
550 225721385084601 0.000063847581 0.001700379866
551 233557522163930 0.000066064112 0.001766443978
552 241627176819381 0.000068346696 0.001834790674
553 249935896266269 0.000070696901 0.001905487575
554 258489317615563 0.000073116323 0.001978603898
555 267293156421037 0.000075606578 0.002054210476
556 276353217726061 0.000078169308 0.002132379785
557 285675384124940 0.000080806178 0.002213185962
558 295265627053394 0.000083518875 0.002296704837
559 305129994336046 0.000086309111 0.002383013949
560 315274621737283 0.000089178622 0.002472192571
561 325705719979621 0.000092129164 0.002564321734
562 336429586543371 0.000095162518 0.002659484253
563 347452592155562 0.000098280487 0.002757764740
564 358781192841696 0.000101484897 0.002859249636
565 370421915855964 0.000104777593 0.002964027229
566 382381372002154 0.000108160446 0.003072187675
567 394666240991089 0.000111635346 0.003183823021
568 407283284019903 0.000115204205 0.003299027226
569 420239328556562 0.000118868953 0.003417896179
570 433541281180913 0.000122631545 0.003540527724
571 447196111767398 0.000126493952 0.003667021677
572 461210866607609 0.000130458168 0.003797479844
573 475592651976889 0.000134526202 0.003932006046
574 490348647526558 0.000138700084 0.004070706130
575 505486089237058 0.000142981863 0.004213687994
576 521012283085036 0.000147373605 0.004361061598
577 536934587353256 0.000151877390 0.004512938988
578 553260426593322 0.000156495319 0.004669434307
579 569997273279789 0.000161229505 0.004830663812
580 587152662058258 0.000166082081 0.004996745893
581 604734170747184 0.000171055189 0.005167801082
582 622749434876858 0.000176150989 0.005343952071
583 641206128010489 0.000181371652 0.005525323723
584 660111976597920 0.000186719363 0.005712043086
585 679474739604064 0.000192196316 0.005904239402
586 699302223666421 0.000197804721 0.006102044123
587 719602262037791 0.000203546793 0.006305590916
588 740382730055707 0.000209424759 0.006515015674
589 761651523370470 0.000215440852 0.006730456527
590 783416573752126 0.000221597318 0.006952053844
591 805685826594439 0.000227896401 0.007179950246
592 828467257050350 0.000234340360 0.007414290606
593 851768846820577 0.000240931451 0.007655222056
594 875598600626154 0.000247671938 0.007902893994
595 899964522255079 0.000254564086 0.008157458080
596 924874631401209 0.000261610163 0.008419068243
597 950336938960611 0.000268812435 0.008687880678
598 976359464227661 0.000276173170 0.008964053847
599 1002950209453776 0.000283694631 0.009247748478
600 1030117177412699 0.000291379084 0.009539127562
601 1057868345195045 0.000299228783 0.009838356345
602 1086211682173956 0.000307245983 0.010145602327
603 1115155123030642 0.000315432928 0.010461035255
604 1144706586113291 0.000323791859 0.010784827115
605 1174873945708909 0.000332325002 0.011117152117
606 1205665050808215 0.000341034579 0.011458186696
607 1237087696599984 0.000349922792 0.011808109488
608 1269149643677458 0.000358991839 0.012167101327
609 1301858588751637 0.000368243896 0.012535345223
610 1335222184292444 0.000377681127 0.012913026350
611 1369248008480895 0.000387305676 0.013300332026
612 1403943585301204 0.000397119672 0.013697451697
613 1439316353708705 0.000407125218 0.014104576915
614 1475373688210195 0.000417324400 0.014521901315
615 1512122867248405 0.000427719278 0.014949620593
616 1549571094266055 0.000438311888 0.015387932481
617 1587725465329523 0.000449104239 0.015837036720
618 1626592990693830 0.000460098312 0.016297135032
619 1666180561645293 0.000471296058 0.016768431090
620 1706494972609528 0.000482699398 0.017251130488
621 1747542887215360 0.000494310216 0.017745440705
622 1789330860941288 0.000506130368 0.018251571073
623 1831865306428571 0.000518161667 0.018769732740
624 1875152516686548 0.000530405893 0.019300138633
625 1919198629635734 0.000542864783 0.019843003416
626 1964009651915660 0.000555540035 0.020398543451
627 2009591422664545 0.000568433300 0.020966976751
628 2055949637927477 0.000581546191 0.021548522943
629 2103089813703016 0.000594880268 0.022143403210
630 2151017310973098 0.000608437046 0.022751840257
631 2199737298000372 0.000622217988 0.023374058245
632 2249254776027773 0.000636224509 0.024010282754
633 2299574540834782 0.000650457964 0.024660740718
634 2350701209103646 0.000664919661 0.025325660379
635 2402639179270463 0.000679610842 0.026005271220
636 2455392658582214 0.000694532698 0.026699803918
637 2508965623226933 0.000709686354 0.027409490272
638 2563361846130743 0.000725072877 0.028134563149
639 2618584856379097 0.000740693265 0.028875256414
640 2674637967752948 0.000756548456 0.029631804870
641 2731524237478957 0.000772639314 0.030404444184
642 2789246495528881 0.000788966639 0.031193410823
643 2847807302685929 0.000805531157 0.031998941980
644 2907208980659856 0.000822333524 0.032821275504
645 2967453569482805 0.000839374316 0.033660649820
646 3028542858437463 0.000856654041 0.034517303861
647 3090478342825046 0.000874173120 0.035391476981
648 3153261255732853 0.000891931903 0.036283408884
649 3216892524160984 0.000909930653 0.037193339537
650 3281372801682291 0.000928169553 0.038121509089
651 3346702423944313 0.000946648698 0.039068157787
652 3412881442219846 0.000965368104 0.040033525892
653 3479909578328192 0.000984327692 0.041017853584
654 3547786259101073 0.001003527299 0.042021380883
655 3616510570712255 0.001022966667 0.043044347549
656 3686081294113980 0.001042645451 0.044086993001
657 3756496858791478 0.001062563207 0.045149556208
658 3827755379165355 0.001082719402 0.046232275610
659 3899854607818599 0.001103113399 0.047335389009
660 3972791972891010 0.001123744472 0.048459133482
661 4046564530766019 0.001144611788 0.049603745270
662 4121169004511073 0.001165714420 0.050769459689
663 4196601736043217 0.001187051332 0.051956511021
664 4272858725610066 0.001208621394 0.053165132416
665 4349935583482269 0.001230423365 0.054395555781
666 4427827570497986 0.001252455903 0.055648011683
667 4506529549270567 0.001274717555 0.056922729239
668 4586036025851501 0.001297206768 0.058219936006
669 4666341100470600 0.001319921872 0.059539857878
670 4747438510307985 0.001342861096 0.060882718974
671 4829321579817147 0.001366022552 0.062248741527
672 4911983264628415 0.001389404248 0.063638145774
673 4995416101440403 0.001413004071 0.065051149845
674 5079612253106599 0.001436819806 0.066487969651
675 5164563458113973 0.001460849115 0.067948818766
676 5250261076840903 0.001485089555 0.069433908322
677 5336696040671980 0.001509538561 0.070943446882
678 5423858899445150 0.001534193459 0.072477640341
679 5511739770187560 0.001559051454 0.074036691794
680 5600328385803359 0.001584109641 0.075620801436
681 5689614043446161 0.001609364994 0.077230166430
682 5779585654429600 0.001634814376 0.078864980806
683 5870231692286645 0.001660454527 0.080525435333
684 5961540243918360 0.001686282076 0.082211717409
685 6053498957320960 0.001712293530 0.083924010940
686 6146095094019128 0.001738485286 0.085662496225
687 6239315476476153 0.001764853616 0.087427349841
688 6333146541790366 0.001791394684 0.089218744525
689 6427574288835366 0.001818104529 0.091036849053
690 6522584333229145 0.001844979083 0.092881828136
691 6618161854183309 0.001872014153 0.094753842289
692 6714291650789160 0.001899205440 0.096653047730
693 6810958088586958 0.001926548522 0.098579596252
694 6908145157139459 0.001954038869 0.100533635121
695 7005836416368480 0.001981671832 0.102515306953
696 7104015055421622 0.002009442655 0.104524749608
697 7202663838749386 0.002037346463 0.106562096071
698 7301765166303075 0.002065378278 0.108627474350
699 7401301019361722 0.002093533003 0.110721007352
700 7501253022027266 0.002121805440 0.112842812792
701 7601602386843987 0.002150190275 0.114993003067
702 7702329977589080 0.002178682095 0.117171685161
703 7803416254655222 0.002207275372 0.119378960534
704 7904841339170873 0.002235964486 0.121614925019
705 8006584958150181 0.002264743699 0.123879668719
706 8108626509899889 0.002293607186 0.126173275905
707 8210945008978195 0.002322549011 0.128495824916
708 8313519152881939 0.002351563147 0.130847388063
709 8416327266778085 0.002380643464 0.133228031527
710 8519347371498567 0.002409783745 0.135637815271
711 8622557128046053 0.002438977670 0.138076792942
712 8725933906847840 0.002468218840 0.140545011781
713 8829454732072484 0.002497500754 0.143042512536
714 8933096352127635 0.002526816837 0.145569329373
715 9036835183748460 0.002556160417 0.148125489790
716 9140647383762528 0.002585524750 0.150711014539
717 9244508792945678 0.002614903002 0.153325917541
718 9348395008996007 0.002644288271 0.155970205812
719 9452281330191897 0.002673673570 0.158643879382
720 9556142829555643 0.002703051847 0.161346931229
721 9659954298267308 0.002732415973 0.164079347203
722 9763690321033408 0.002761758759 0.166841105962
723 9867325219249981 0.002791072940 0.169632178902
724 9970833127512512 0.002820351201 0.172452530103
725 10074187936558122 0.002849586156 0.175302116259
726 10177363370888455 0.002878770373 0.178180886632
727 10280332931441733 0.002907896357 0.181088782990
728 10383069974340297 0.002936956571 0.184025739561
729 10485547653279647 0.002965943421 0.186991682982
730 10587738999328217 0.002994849279 0.189986532261
731 10689616863061737 0.003023666465 0.193010198726
732 10791153995382729 0.003052387272 0.196062585998
733 10892322989344543 0.003081003947 0.199143589945
734 10993096361994634 0.003109508717 0.202253098662
735 11093446495874170 0.003137893770 0.205390992432
736 11193345721807668 0.003166151278 0.208557143710
737 11292766260101878 0.003194273385 0.211751417095
738 11391680304241666 0.003222252225 0.214973669319
739 11490059961730183 0.003250079908 0.218223749227
740 11587877338692722 0.003277748544 0.221501497772
741 11685104480338701 0.003305250227 0.224806747998
742 11781713456388257 0.003332577055 0.228139325053
743 11877676301179640 0.003359721118 0.231499046171
744 11972965099875122 0.003386674520 0.234885720691
745 12067551928151070 0.003413429363 0.238299150054
746 12161408939178544 0.003439977770 0.241739127824
747 12254508302876326 0.003466311869 0.245205439693
748 12346822293578054 0.003492423816 0.248697863509
749 12438323228868060 0.003518305783 0.252216169292
750 12528983557886059 0.003543949976 0.255760119269
751 12618775799682379 0.003569348622 0.259329467890
752 12707672632154594 0.003594493992 0.262923961882
753 12795646829899064 0.003619378385 0.266543340267
754 12882671353685064 0.003643994154 0.270187334421
755 12968719287824486 0.003668333684 0.273855668105
756 13053763930133222 0.003692389423 0.277548057528
757 13137778728752338 0.003716153860 0.281264211388
758 13220737372587114 0.003739619553 0.285003830941
759 13302613727557896 0.003762779110 0.288766610051
760 13383381927422176 0.003785625214 0.292552235265
761 13463016309475063 0.003808150606 0.296360385871
762 13541491505700427 0.003830348111 0.300190733982
763 13618782377856223 0.003852210617 0.304042944599
764 13694864108945970 0.003873731106 0.307916675705
765 13769712137666260 0.003894902629 0.311811578334
766 13843302250100143 0.003915718338 0.315727296673
767 13915610513550433 0.003936171464 0.319663468136
768 13986613368402674 0.003956255341 0.323619723477
769 14056287561278035 0.003975963392 0.327595686869
770 14124610237055954 0.003995289153 0.331590976022
771 14191558871327486 0.004014226253 0.335605202276
772 14257111362481292 0.004032768440 0.339637970715
773 14321245963481667 0.004050909562 0.343688880277
774 14383941373963924 0.004068643594 0.347757523871
775 14445176671275633 0.004085964618 0.351843488489
776 14504931402574878 0.004102866850 0.355946355338
777 14563185515116493 0.004119344616 0.360065699954
778 14619919448256126 0.004135392384 0.364201092338
779 14675114063011110 0.004151004740 0.368352097077
780 14728750733921081 0.004166176416 0.372518273493
781 14780811277822859 0.004180902268 0.376699175761
782 14831278045561927 0.004195177305 0.380894353066
783 14880133849969280 0.004208996664 0.385103349731
784 14927362057331041 0.004222355641 0.389325705372
785 14972946514599961 0.004235249667 0.393560955039
786 15016871640574771 0.004247674334 0.397808629373
787 15059122352292240 0.004259625376 0.402068254750
788 15099684155916740 0.004271098693 0.406339353443
789 15138543072303315 0.004282090332 0.410621443775
790 15175685727506919 0.004292596508 0.414914040283
791 15211099277556439 0.004302613590 0.419216653873
792 15244771498541818 0.004312138119 0.423528791992
793 15276690710548182 0.004321166791 0.427849958783
794 15306845867323822 0.004329696483 0.432179655266
795 15335226479373283 0.004337724227 0.436517379493
796 15361822703122288 0.004345247238 0.440862626731
797 15386625263213894 0.004352262894 0.445214889625
798 15409625541131887 0.004358768755 0.449573658380
799 15430815496656959 0.004364762549 0.453938420929
800 15450187755959996 0.004370242189 0.458308663118
801 15467735532221058 0.004375205755 0.462683868873
802 15483452713112335 0.004379651519 0.467063520392
803 15497333780632956 0.004383577920 0.471447098312
804 15509373897951036 0.004386983589 0.475834081901
805 15519568828416239 0.004389867328 0.480223949230
806 15527915021776296 0.004392228134 0.484616177364
807 15534409532379142 0.004394065172 0.489010242536
808 15539050104692302 0.004395377805 0.493405620340
809 15541835090752144 0.004396165566 0.497801785907
810 15542763534960598 0.004396428186 0.502198214093
811 15541835090752144 0.004396165566 0.506594379660
812 15539050104692302 0.004395377805 0.510989757464
813 15534409532379142 0.004394065172 0.515383822636
814 15527915021776296 0.004392228134 0.519776050770
815 15519568828416239 0.004389867328 0.524165918099
816 15509373897951036 0.004386983589 0.528552901688
817 15497333780632956 0.004383577920 0.532936479608
818 15483452713112335 0.004379651519 0.537316131127
819 15467735532221058 0.004375205755 0.541691336882
820 15450187755959996 0.004370242189 0.546061579071
821 15430815496656959 0.004364762549 0.550426341620
822 15409625541131887 0.004358768755 0.554785110375
823 15386625263213894 0.004352262894 0.559137373269
824 15361822703122288 0.004345247238 0.563482620507
825 15335226479373283 0.004337724227 0.567820344734
826 15306845867323822 0.004329696483 0.572150041217
827 15276690710548182 0.004321166791 0.576471208008
828 15244771498541818 0.004312138119 0.580783346127
829 15211099277556439 0.004302613590 0.585085959717
830 15175685727506919 0.004292596508 0.589378556225
831 15138543072303315 0.004282090332 0.593660646557
832 15099684155916740 0.004271098693 0.597931745250
833 15059122352292240 0.004259625376 0.602191370627
834 15016871640574771 0.004247674334 0.606439044961
835 14972946514599961 0.004235249667 0.610674294628
836 14927362057331041 0.004222355641 0.614896650269
837 14880133849969280 0.004208996664 0.619105646934
838 14831278045561927 0.004195177305 0.623300824239
839 14780811277822859 0.004180902268 0.627481726507
840 14728750733921081 0.004166176416 0.631647902923
841 14675114063011110 0.004151004740 0.635798907662
842 14619919448256126 0.004135392384 0.639934300046
843 14563185515116493 0.004119344616 0.644053644662
844 14504931402574878 0.004102866850 0.648156511511
845 14445176671275633 0.004085964618 0.652242476129
846 14383941373963924 0.004068643594 0.656311119723
847 14321245963481667 0.004050909562 0.660362029285
848 14257111362481292 0.004032768440 0.664394797724
849 14191558871327486 0.004014226253 0.668409023978
850 14124610237055954 0.003995289153 0.672404313131
851 14056287561278035 0.003975963392 0.676380276523
852 13986613368402674 0.003956255341 0.680336531864
853 13915610513550433 0.003936171464 0.684272703327
854 13843302250100143 0.003915718338 0.688188421666
855 13769712137666260 0.003894902629 0.692083324295
856 13694864108945970 0.003873731106 0.695957055401
857 13618782377856223 0.003852210617 0.699809266018
858 13541491505700427 0.003830348111 0.703639614129
859 13463016309475063 0.003808150606 0.707447764735
860 13383381927422176 0.003785625214 0.711233389949
861 13302613727557896 0.003762779110 0.714996169059
862 13220737372587114 0.003739619553 0.718735788612
863 13137778728752338 0.003716153860 0.722451942472
864 13053763930133222 0.003692389423 0.726144331895
865 12968719287824486 0.003668333684 0.729812665579
866 12882671353685064 0.003643994154 0.733456659733
867 12795646829899064 0.003619378385 0.737076038118
868 12707672632154594 0.003594493992 0.740670532110
869 12618775799682379 0.003569348622 0.744239880731
870 12528983557886059 0.003543949976 0.747783830708
871 12438323228868060 0.003518305783 0.751302136491
872 12346822293578054 0.003492423816 0.754794560307
873 12254508302876326 0.003466311869 0.758260872176
874 12161408939178544 0.003439977770 0.761700849946
875 12067551928151070 0.003413429363 0.765114279309
876 11972965099875122 0.003386674520 0.768500953829
877 11877676301179640 0.003359721118 0.771860674947
878 11781713456388257 0.003332577055 0.775193252002
879 11685104480338701 0.003305250227 0.778498502228
880 11587877338692722 0.003277748544 0.781776250773
881 11490059961730183 0.003250079908 0.785026330681
882 11391680304241666 0.003222252225 0.788248582905
883 11292766260101878 0.003194273385 0.791442856290
884 11193345721807668 0.003166151278 0.794609007568
885 11093446495874170 0.003137893770 0.797746901338
886 10993096361994634 0.003109508717 0.800856410055
887 10892322989344543 0.003081003947 0.803937414002
888 10791153995382729 0.003052387272 0.806989801274
889 10689616863061737 0.003023666465 0.810013467739
890 10587738999328217 0.002994849279 0.813008317018
891 10485547653279647 0.002965943421 0.815974260439
892 10383069974340297 0.002936956571 0.818911217010
893 10280332931441733 0.002907896357 0.821819113368
894 10177363370888455 0.002878770373 0.824697883741
895 10074187936558122 0.002849586156 0.827547469897
896 9970833127512512 0.002820351201 0.830367821098
897 9867325219249981 0.002791072940 0.833158894038
898 9763690321033408 0.002761758759 0.835920652797
899 9659954298267308 0.002732415973 0.838653068771
900 9556142829555643 0.002703051847 0.841356120618
901 9452281330191897 0.002673673570 0.844029794188
902 9348395008996007 0.002644288271 0.846674082459
903 9244508792945678 0.002614903002 0.849288985461
904 9140647383762528 0.002585524750 0.851874510210
905 9036835183748460 0.002556160417 0.854430670627
906 8933096352127635 0.002526816837 0.856957487464
907 8829454732072484 0.002497500754 0.859454988219
908 8725933906847840 0.002468218840 0.861923207058
909 8622557128046053 0.002438977670 0.864362184729
910 8519347371498567 0.002409783745 0.866771968473
911 8416327266778085 0.002380643464 0.869152611937
912 8313519152881939 0.002351563147 0.871504175084
913 8210945008978195 0.002322549011 0.873826724095
914 8108626509899889 0.002293607186 0.876120331281
915 8006584958150181 0.002264743699 0.878385074981
916 7904841339170873 0.002235964486 0.880621039466
917 7803416254655222 0.002207275372 0.882828314839
918 7702329977589080 0.002178682095 0.885006996933
919 7601602386843987 0.002150190275 0.887157187208
920 7501253022027266 0.002121805440 0.889278992648
921 7401301019361722 0.002093533003 0.891372525650
922 7301765166303075 0.002065378278 0.893437903929
923 7202663838749386 0.002037346463 0.895475250392
924 7104015055421622 0.002009442655 0.897484693047
925 7005836416368480 0.001981671832 0.899466364879
926 6908145157139459 0.001954038869 0.901420403748
927 6810958088586958 0.001926548522 0.903346952270
928 6714291650789160 0.001899205440 0.905246157711
929 6618161854183309 0.001872014153 0.907118171864
930 6522584333229145 0.001844979083 0.908963150947
931 6427574288835366 0.001818104529 0.910781255475
932 6333146541790366 0.001791394684 0.912572650159
933 6239315476476153 0.001764853616 0.914337503775
934 6146095094019128 0.001738485286 0.916075989060
935 6053498957320960 0.001712293530 0.917788282591
936 5961540243918360 0.001686282076 0.919474564667
937 5870231692286645 0.001660454527 0.921135019194
938 5779585654429600 0.001634814376 0.922769833570
939 5689614043446161 0.001609364994 0.924379198564
940 5600328385803359 0.001584109641 0.925963308206
941 5511739770187560 0.001559051454 0.927522359659
942 5423858899445150 0.001534193459 0.929056553118
943 5336696040671980 0.001509538561 0.930566091678
944 5250261076840903 0.001485089555 0.932051181234
945 5164563458113973 0.001460849115 0.933512030349
946 5079612253106599 0.001436819806 0.934948850155
947 4995416101440403 0.001413004071 0.936361854226
948 4911983264628415 0.001389404248 0.937751258473
949 4829321579817147 0.001366022552 0.939117281026
950 4747438510307985 0.001342861096 0.940460142122
951 4666341100470600 0.001319921872 0.941780063994
952 4586036025851501 0.001297206768 0.943077270761
953 4506529549270567 0.001274717555 0.944351988317
954 4427827570497986 0.001252455903 0.945604444219
955 4349935583482269 0.001230423365 0.946834867584
956 4272858725610066 0.001208621394 0.948043488979
957 4196601736043217 0.001187051332 0.949230540311
958 4121169004511073 0.001165714420 0.950396254730
959 4046564530766019 0.001144611788 0.951540866518
960 3972791972891010 0.001123744472 0.952664610991
961 3899854607818599 0.001103113399 0.953767724390
962 3827755379165355 0.001082719402 0.954850443792
963 3756496858791478 0.001062563207 0.955913006999
964 3686081294113980 0.001042645451 0.956955652451
965 3616510570712255 0.001022966667 0.957978619117
966 3547786259101073 0.001003527299 0.958982146416
967 3479909578328192 0.000984327692 0.959966474108
968 3412881442219846 0.000965368104 0.960931842213
969 3346702423944313 0.000946648698 0.961878490911
970 3281372801682291 0.000928169553 0.962806660463
971 3216892524160984 0.000909930653 0.963716591116
972 3153261255732853 0.000891931903 0.964608523019
973 3090478342825046 0.000874173120 0.965482696139
974 3028542858437463 0.000856654041 0.966339350180
975 2967453569482805 0.000839374316 0.967178724496
976 2907208980659856 0.000822333524 0.968001058020
977 2847807302685929 0.000805531157 0.968806589177
978 2789246495528881 0.000788966639 0.969595555816
979 2731524237478957 0.000772639314 0.970368195130
980 2674637967752948 0.000756548456 0.971124743586
981 2618584856379097 0.000740693265 0.971865436851
982 2563361846130743 0.000725072877 0.972590509728
983 2508965623226933 0.000709686354 0.973300196082
984 2455392658582214 0.000694532698 0.973994728780
985 2402639179270463 0.000679610842 0.974674339621
986 2350701209103646 0.000664919661 0.975339259282
987 2299574540834782 0.000650457964 0.975989717246
988 2249254776027773 0.000636224509 0.976625941755
989 2199737298000372 0.000622217988 0.977248159743
990 2151017310973098 0.000608437046 0.977856596790
991 2103089813703016 0.000594880268 0.978451477057
992 2055949637927477 0.000581546191 0.979033023249
993 2009591422664545 0.000568433300 0.979601456549
994 1964009651915660 0.000555540035 0.980156996584
995 1919198629635734 0.000542864783 0.980699861367
996 1875152516686548 0.000530405893 0.981230267260
997 1831865306428571 0.000518161667 0.981748428927
998 1789330860941288 0.000506130368 0.982254559295
999 1747542887215360 0.000494310216 0.982748869512
1000 1706494972609528 0.000482699398 0.983231568910
1001 1666180561645293 0.000471296058 0.983702864968
1002 1626592990693830 0.000460098312 0.984162963280
1003 1587725465329523 0.000449104239 0.984612067519
1004 1549571094266055 0.000438311888 0.985050379407
1005 1512122867248405 0.000427719278 0.985478098685
1006 1475373688210195 0.000417324400 0.985895423085
1007 1439316353708705 0.000407125218 0.986302548303
1008 1403943585301204 0.000397119672 0.986699667974
1009 1369248008480895 0.000387305676 0.987086973650
1010 1335222184292444 0.000377681127 0.987464654777
1011 1301858588751637 0.000368243896 0.987832898673
1012 1269149643677458 0.000358991839 0.988191890512
1013 1237087696599984 0.000349922792 0.988541813304
1014 1205665050808215 0.000341034579 0.988882847883
1015 1174873945708909 0.000332325002 0.989215172885
1016 1144706586113291 0.000323791859 0.989538964745
1017 1115155123030642 0.000315432928 0.989854397673
1018 1086211682173956 0.000307245983 0.990161643655
1019 1057868345195045 0.000299228783 0.990460872438
1020 1030117177412699 0.000291379084 0.990752251522
1021 1002950209453776 0.000283694631 0.991035946153
1022 976359464227661 0.000276173170 0.991312119322
1023 950336938960611 0.000268812435 0.991580931757
1024 924874631401209 0.000261610163 0.991842541920
1025 899964522255079 0.000254564086 0.992097106006
1026 875598600626154 0.000247671938 0.992344777944
1027 851768846820577 0.000240931451 0.992585709394
1028 828467257050350 0.000234340360 0.992820049754
1029 805685826594439 0.000227896401 0.993047946156
1030 783416573752126 0.000221597318 0.993269543473
1031 761651523370470 0.000215440852 0.993484984326
1032 740382730055707 0.000209424759 0.993694409084
1033 719602262037791 0.000203546793 0.993897955877
1034 699302223666421 0.000197804721 0.994095760598
1035 679474739604064 0.000192196316 0.994287956914
1036 660111976597920 0.000186719363 0.994474676277
1037 641206128010489 0.000181371652 0.994656047929
1038 622749434876858 0.000176150989 0.994832198918
1039 604734170747184 0.000171055189 0.995003254107
1040 587152662058258 0.000166082081 0.995169336188
1041 569997273279789 0.000161229505 0.995330565693
1042 553260426593322 0.000156495319 0.995487061012
1043 536934587353256 0.000151877390 0.995638938402
1044 521012283085036 0.000147373605 0.995786312006
1045 505486089237058 0.000142981863 0.995929293870
1046 490348647526558 0.000138700084 0.996067993954
1047 475592651976889 0.000134526202 0.996202520156
1048 461210866607609 0.000130458168 0.996332978323
1049 447196111767398 0.000126493952 0.996459472276
1050 433541281180913 0.000122631545 0.996582103821
1051 420239328556562 0.000118868953 0.996700972774
1052 407283284019903 0.000115204205 0.996816176979
1053 394666240991089 0.000111635346 0.996927812325
1054 382381372002154 0.000108160446 0.997035972771
1055 370421915855964 0.000104777593 0.997140750364
1056 358781192841696 0.000101484897 0.997242235260
1057 347452592155562 0.000098280487 0.997340515747
1058 336429586543371 0.000095162518 0.997435678266
1059 325705719979621 0.000092129164 0.997527807429
1060 315274621737283 0.000089178622 0.997616986051
1061 305129994336046 0.000086309111 0.997703295163
1062 295265627053394 0.000083518875 0.997786814038
1063 285675384124940 0.000080806178 0.997867620215
1064 276353217726061 0.000078169308 0.997945789524
1065 267293156421037 0.000075606578 0.998021396102
1066 258489317615563 0.000073116323 0.998094512425
1067 249935896266269 0.000070696901 0.998165209326
1068 241627176819381 0.000068346696 0.998233556022
1069 233557522163930 0.000066064112 0.998299620134
1070 225721385084601 0.000063847581 0.998363467716
1071 218113297456569 0.000061695557 0.998425163272
1072 210727881214639 0.000059606517 0.998484769789
1073 203559837800120 0.000057578963 0.998542348752
1074 196603958660512 0.000055611422 0.998597960175
1075 189855114934011 0.000053702443 0.998651662618
1076 183308267507535 0.000051850601 0.998703513219
1077 176958456936318 0.000050054493 0.998753567712
1078 170800813062207 0.000048312741 0.998801880453
1079 164830545178892 0.000046623990 0.998848504443
1080 159042951225485 0.000044986911 0.998893491354
1081 153433408182971 0.000043400194 0.998936891548
1082 147997380868998 0.000041862559 0.998978754107
1083 142730412563985 0.000040372744 0.999019126851
1084 137628133409981 0.000038929512 0.999058056364
1085 132686251275763 0.000037531651 0.999095588014
1086 127900559773363 0.000036177970 0.999131765984
1087 123266929349329 0.000034867300 0.999166633284
1088 118781314944316 0.000033598499 0.999200231783
1089 114439747308027 0.000032370442 0.999232602225
1090 110238340304328 0.000031182032 0.999263784257
1091 106173282459017 0.000030032189 0.999293816446
1092 102240843924128 0.000028919859 0.999322736305
1093 98437368245632 0.000027844007 0.999350580312
1094 94759279010024 0.000026803622 0.999377383935
1095 91203071830294 0.000025797713 0.999403181648
1096 87765320677722 0.000024825310 0.999428006958
1097 84442670093739 0.000023885465 0.999451892423
1098 81231841219122 0.000022977250 0.999474869673
1099 78129624219915 0.000022099756 0.999496969429
1100 75132884036461 0.000021252098 0.999518221527
1101 72238553020681 0.000020433407 0.999538654933
1102 69443636407019 0.000019642836 0.999558297770
1103 66745205168528 0.000018879558 0.999577177327
1104 64140401221500 0.000018142763 0.999595320090
1105 61626430488492 0.000017431660 0.999612751750
1106 59200567856702 0.000016745481 0.999629497231
1107 56860150445667 0.000016083470 0.999645580701
1108 54602582321801 0.000015444894 0.999661025595
1109 52425327976532 0.000014829035 0.999675854630
1110 50325916807381 0.000014235196 0.999690089825
1111 48301936795528 0.000013662692 0.999703752518
1112 46351038772509 0.000013110861 0.999716863379
1113 44470930294083 0.000012579053 0.999729442432
1114 42659379693854 0.000012066638 0.999741509070
1115 40914210159126 0.000011572999 0.999753082069
1116 39233303581272 0.000011097538 0.999764179607
1117 37614594822327 0.000010639669 0.999774819276
1118 36056075379039 0.000010198826 0.999785018102
1119 34555787837266 0.000009774455 0.999794792557
1120 33111829351342 0.000009366017 0.999804158574
1121 31722346290416 0.000008972987 0.999813131561
1122 30385537541434 0.000008594857 0.999821726418
1123 29099649337087 0.000008231131 0.999829957549
1124 27862978398196 0.000007881326 0.999837838875
1125 26673866939713 0.000007544974 0.999845383848
1126 25530705653070 0.000007221619 0.999852605467
1127 24431928894154 0.000006910819 0.999859516286
1128 23376017513498 0.000006612143 0.999866128429
1129 22361494215565 0.000006325175 0.999872453604
1130 21386926250737 0.000006049509 0.999878503113
1131 20450920942555 0.000005784750 0.999884287864
1132 19552128242181 0.000005530518 0.999889818381
1133 18689236426307 0.000005286440 0.999895104821
1134 17860974520443 0.000005052158 0.999900156979
1135 17066108157849 0.000004827322 0.999904984300
1136 16303441884554 0.000004611594 0.999909595894
1137 15571815175325 0.000004404646 0.999914000540
1138 14870104620270 0.000004206160 0.999918206700
1139 14197220100303 0.000004015828 0.999922222527
1140 13552106861442 0.000003833351 0.999926055878
1141 12933741839899 0.000003658440 0.999929714318
1142 12341135635235 0.000003490815 0.999933205133
1143 11773328981483 0.000003330205 0.999936535339
1144 11229394619008 0.000003176348 0.999939711687
1145 10708433913262 0.000003028989 0.999942740676
1146 10209578630140 0.000002887883 0.999945628559
1147 9731987693363 0.000002752791 0.999948381350
1148 9274848873719 0.000002623485 0.999951004835
1149 8837375680898 0.000002499741 0.999953504576
1150 8418808965914 0.000002381345 0.999955885922
1151 8018413948848 0.000002268090 0.999958154011
1152 7635481738958 0.000002159773 0.999960313785
1153 7269326489361 0.000002056203 0.999962369987
1154 6919286843610 0.000001957190 0.999964327178
1155 6584723213704 0.000001862556 0.999966189734
1156 6265019152487 0.000001772124 0.999967961858
1157 5959578755669 0.000001685727 0.999969647585
1158 5667827963566 0.000001603203 0.999971250788
1159 5389212078927 0.000001524393 0.999972775181
1160 5123197006289 0.000001449148 0.999974224329
1161 4869266881582 0.000001377321 0.999975601650
1162 4626925247931 0.000001308773 0.999976910423
1163 4395692797816 0.000001243366 0.999978153789
1164 4175108488550 0.000001180972 0.999979334761
1165 3964727389027 0.000001121463 0.999980456224
1166 3764121741811 0.000001064720 0.999981520944
1167 3572878910940 0.000001010625 0.999982531569
1168 3390602389686 0.000000959066 0.999983490636
1169 3216909849508 0.000000909936 0.999984400571
1170 3051434096031 0.000000863129 0.999985263700
1171 2893821211946 0.000000818547 0.999986082247
1172 2743731467669 0.000000776092 0.999986858339
1173 2600837554495 0.000000735673 0.999987594012
1174 2464825448513 0.000000697201 0.999988291213
1175 2335392734382 0.000000660590 0.999988951803
1176 2212249424868 0.000000625757 0.999989577560
1177 2095116368246 0.000000592625 0.999990170185
1178 1983726028916 0.000000561117 0.999990731302
1179 1877820975168 0.000000531161 0.999991262463
1180 1777154619819 0.000000502686 0.999991765149
1181 1681489788194 0.000000475626 0.999992240775
1182 1590599420536 0.000000449917 0.999992690692
1183 1504265214046 0.000000425497 0.999993116189
1184 1422278292091 0.000000402306 0.999993518495
1185 1344437917067 0.000000380288 0.999993898783
1186 1270552124993 0.000000359389 0.999994258171
1187 1200436509276 0.000000339556 0.999994597727
1188 1133914822434 0.000000320739 0.999994918466
1189 1070817824773 0.000000302892 0.999995221358
1190 1010983857544 0.000000285967 0.999995507325
1191 954257753733 0.000000269921 0.999995777246
1192 900491381529 0.000000254713 0.999996031960
1193 849542617012 0.000000240302 0.999996272261
1194 801275859118 0.000000226649 0.999996498910
1195 755561059108 0.000000213718 0.999996712628
1196 712274211009 0.000000201474 0.999996914102
1197 671296435063 0.000000189883 0.999997103985
1198 632514442353 0.000000178913 0.999997282899
1199 595819672189 0.000000168534 0.999997451432
1200 561108732221 0.000000158715 0.999997610147
1201 528282584833 0.000000149430 0.999997759578
1202 497246966079 0.000000140651 0.999997900229
1203 467911618783 0.000000132354 0.999998032582
1204 440190689221 0.000000124512 0.999998157095
1205 414002006749 0.000000117105 0.999998274199
1206 389267459099 0.000000110108 0.999998384308
1207 365912314318 0.000000103502 0.999998487810
1208 343865577844 0.000000097266 0.999998585076
1209 323059354425 0.000000091381 0.999998676456
1210 303429185857 0.000000085828 0.999998762284
1211 284913452879 0.000000080591 0.999998842875
1212 267453694512 0.000000075652 0.999998918527
1213 250994045973 0.000000070996 0.999998989523
1214 235481542190 0.000000066608 0.999999056131
1215 220865589927 0.000000062474 0.999999118605
1216 207098254570 0.000000058580 0.999999177185
1217 194133766301 0.000000054913 0.999999232098
1218 181928790813 0.000000051460 0.999999283558
1219 170441966142 0.000000048211 0.999999331770
1220 159634159836 0.000000045154 0.999999376924
1221 149468034620 0.000000042279 0.999999419202
1222 139908291043 0.000000039574 0.999999458777
1223 130921262045 0.000000037032 0.999999495809
1224 122475141756 0.000000034643 0.999999530453
1225 114539605867 0.000000032399 0.999999562851
1226 107086028628 0.000000030290 0.999999593142
1227 100087127594 0.000000028311 0.999999621452
1228 93517168095 0.000000026452 0.999999647905
1229 87351632249 0.000000024708 0.999999672613
1230 81567411419 0.000000023072 0.999999695685
1231 76142496873 0.000000021538 0.999999717223
1232 71056162189 0.000000020099 0.999999737322
1233 66288674187 0.000000018750 0.999999756072
1234 61821464449 0.000000017487 0.999999773559
1235 57636860670 0.000000016303 0.999999789862
1236 53718247833 0.000000015195 0.999999805057
1237 50049817550 0.000000014157 0.999999819214
1238 46616720518 0.000000013186 0.999999832400
1239 43404832757 0.000000012277 0.999999844677
1240 40400898786 0.000000011428 0.999999856105
1241 37592314751 0.000000010633 0.999999866739
1242 34967262684 0.000000009891 0.999999876629
1243 32514508546 0.000000009197 0.999999885827
1244 30223529082 0.000000008549 0.999999894376
1245 28084323773 0.000000007944 0.999999902319
1246 26087533643 0.000000007379 0.999999909699
1247 24224267278 0.000000006852 0.999999916551
1248 22486212019 0.000000006360 0.999999922911
1249 20865472184 0.000000005902 0.999999928813
1250 19354673930 0.000000005475 0.999999934288
1251 17946814901 0.000000005076 0.999999939364
1252 16635362300 0.000000004705 0.999999944070
1253 15414114032 0.000000004360 0.999999948430
1254 14277290236 0.000000004038 0.999999952468
1255 13219404469 0.000000003739 0.999999956207
1256 12235349889 0.000000003461 0.999999959668
1257 11320279729 0.000000003202 0.999999962870
1258 10469687646 0.000000002961 0.999999965832
1259 9679297643 0.000000002738 0.999999968570
1260 8945138936 0.000000002530 0.999999971100
1261 8263443950 0.000000002337 0.999999973437
1262 7630718671 0.000000002158 0.999999975596
1263 7043648186 0.000000001992 0.999999977588
1264 6499162145 0.000000001838 0.999999979427
1265 5994347965 0.000000001696 0.999999981122
1266 5526511602 0.000000001563 0.999999982685
1267 5093097335 0.000000001441 0.999999984126
1268 4691744778 0.000000001327 0.999999985453
1269 4320214692 0.000000001222 0.999999986675
1270 3976441876 0.000000001125 0.999999987800
1271 3658467191 0.000000001035 0.999999988835
1272 3364486581 0.000000000952 0.999999989786
1273 3092788314 0.000000000875 0.999999990661
1274 2841798846 0.000000000804 0.999999991465
1275 2610024936 0.000000000738 0.999999992203
1276 2396096082 0.000000000678 0.999999992881
1277 2198711618 0.000000000622 0.999999993503
1278 2016679881 0.000000000570 0.999999994073
1279 1848869496 0.000000000523 0.999999994596
1280 1694245990 0.000000000479 0.999999995076
1281 1551826889 0.000000000439 0.999999995515
1282 1420715489 0.000000000402 0.999999995916
1283 1300059954 0.000000000368 0.999999996284
1284 1189084411 0.000000000336 0.999999996621
1285 1087051347 0.000000000307 0.999999996928
1286 993290571 0.000000000281 0.999999997209
1287 907164667 0.000000000257 0.999999997466
1288 828095629 0.000000000234 0.999999997700
1289 755533468 0.000000000214 0.999999997914
1290 688980643 0.000000000195 0.999999998108
1291 627963266 0.000000000178 0.999999998286
1292 572053851 0.000000000162 0.999999998448
1293 520844871 0.000000000147 0.999999998595
1294 473969587 0.000000000134 0.999999998729
1295 431078138 0.000000000122 0.999999998851
1296 391856581 0.000000000111 0.999999998962
1297 356005000 0.000000000101 0.999999999063
1298 323255164 0.000000000091 0.999999999154
1299 293350487 0.000000000083 0.999999999237
1300 266062168 0.000000000075 0.999999999312
1301 241171101 0.000000000068 0.999999999381
1302 218482569 0.000000000062 0.999999999442
1303 197809729 0.000000000056 0.999999999498
1304 178987225 0.000000000051 0.999999999549
1305 161856093 0.000000000046 0.999999999595
1306 146276124 0.000000000041 0.999999999636
1307 132112328 0.000000000037 0.999999999674
1308 119246146 0.000000000034 0.999999999707
1309 107563103 0.000000000030 0.999999999738
1310 96963169 0.000000000027 0.999999999765
1311 87349501 0.000000000025 0.999999999790
1312 78637842 0.000000000022 0.999999999812
1313 70746444 0.000000000020 0.999999999832
1314 63604540 0.000000000018 0.999999999850
1315 57143201 0.000000000016 0.999999999866
1316 51303148 0.000000000015 0.999999999881
1317 46026431 0.000000000013 0.999999999894
1318 41263462 0.000000000012 0.999999999905
1319 36965620 0.000000000010 0.999999999916
1320 33091578 0.000000000009 0.999999999925
1321 29600587 0.000000000008 0.999999999934
1322 26458297 0.000000000007 0.999999999941
1323 23630664 0.000000000007 0.999999999948
1324 21089176 0.000000000006 0.999999999954
1325 18805464 0.000000000005 0.999999999959
1326 16755957 0.000000000005 0.999999999964
1327 14917024 0.000000000004 0.999999999968
1328 13269250 0.000000000004 0.999999999972
1329 11793028 0.000000000003 0.999999999975
1330 10472375 0.000000000003 0.999999999978
1331 9291060 0.000000000003 0.999999999981
1332 8236001 0.000000000002 0.999999999983
1333 7293767 0.000000000002 0.999999999985
1334 6453684 0.000000000002 0.999999999987
1335 5704686 0.000000000002 0.999999999989
1336 5038070 0.000000000001 0.999999999990
1337 4444748 0.000000000001 0.999999999991
1338 3917670 0.000000000001 0.999999999992
1339 3449359 0.000000000001 0.999999999993
1340 3034135 0.000000000001 0.999999999994
1341 2665885 0.000000000001 0.999999999995
1342 2340024 0.000000000001 0.999999999996
1343 2051569 0.000000000001 0.999999999996
1344 1796855 0.000000000001 0.999999999997
1345 1571812 0.000000000000 0.999999999997
1346 1373524 0.000000000000 0.999999999997
1347 1198689 0.000000000000 0.999999999998
1348 1044984 0.000000000000 0.999999999998
1349 909741 0.000000000000 0.999999999998
1350 791131 0.000000000000 0.999999999999
1351 686983 0.000000000000 0.999999999999
1352 595872 0.000000000000 0.999999999999
1353 516054 0.000000000000 0.999999999999
1354 446405 0.000000000000 0.999999999999
1355 385528 0.000000000000 0.999999999999
1356 332557 0.000000000000 0.999999999999
1357 286364 0.000000000000 1.000000000000
1358 246288 0.000000000000 1.000000000000
1359 211428 0.000000000000 1.000000000000
1360 181274 0.000000000000 1.000000000000
1361 155112 0.000000000000 1.000000000000
1362 132559 0.000000000000 1.000000000000
1363 113039 0.000000000000 1.000000000000
1364 96271 0.000000000000 1.000000000000
1365 81801 0.000000000000 1.000000000000
1366 69414 0.000000000000 1.000000000000
1367 58754 0.000000000000 1.000000000000
1368 49668 0.000000000000 1.000000000000
1369 41869 0.000000000000 1.000000000000
1370 35251 0.000000000000 1.000000000000
1371 29588 0.000000000000 1.000000000000
1372 24803 0.000000000000 1.000000000000
1373 20722 0.000000000000 1.000000000000
1374 17293 0.000000000000 1.000000000000
1375 14375 0.000000000000 1.000000000000
1376 11937 0.000000000000 1.000000000000
1377 9871 0.000000000000 1.000000000000
1378 8154 0.000000000000 1.000000000000
1379 6703 0.000000000000 1.000000000000
1380 5507 0.000000000000 1.000000000000
1381 4498 0.000000000000 1.000000000000
1382 3673 0.000000000000 1.000000000000
1383 2980 0.000000000000 1.000000000000
1384 2417 0.000000000000 1.000000000000
1385 1946 0.000000000000 1.000000000000
1386 1568 0.000000000000 1.000000000000
1387 1251 0.000000000000 1.000000000000
1388 1000 0.000000000000 1.000000000000
1389 791 0.000000000000 1.000000000000
1390 627 0.000000000000 1.000000000000
1391 490 0.000000000000 1.000000000000
1392 385 0.000000000000 1.000000000000
1393 297 0.000000000000 1.000000000000
1394 231 0.000000000000 1.000000000000
1395 176 0.000000000000 1.000000000000
1396 135 0.000000000000 1.000000000000
1397 101 0.000000000000 1.000000000000
1398 77 0.000000000000 1.000000000000
1399 56 0.000000000000 1.000000000000
1400 42 0.000000000000 1.000000000000
1401 30 0.000000000000 1.000000000000
1402 22 0.000000000000 1.000000000000
1403 15 0.000000000000 1.000000000000
1404 11 0.000000000000 1.000000000000
1405 7 0.000000000000 1.000000000000
1406 5 0.000000000000 1.000000000000
1407 3 0.000000000000 1.000000000000
1408 2 0.000000000000 1.000000000000
1409 1 0.000000000000 1.000000000000
1410 1 0.000000000000 1.000000000000
Total 3,535,316,142,212,174,320 1.000000000000

The average ball is 40.5, so the average sum is 20×40.5=810. As the table shows, the probability of getting exactly 810, which the mean, median, and mode, is 0.44%, or 1 in 227.46.

## Programming Tips

My first attempt to get the combinations for any given total was by brute force, looping through all 3,535,316,142,212,174,320 ways to choose 20 balls out of 80 and taking a count of the sum of each. That program would have taken years to complete.

My second attempt was to look for a pattern. Perhaps a variant of the Fibonacci Sequence or line from a modified Pascal's Triangle. That was a noble attempt, but led to nothing.

My third attempt was at finding the generating function, but that would have been just as tedious as the first method. This can be done very easily in Mathematica, by the way, but I'm too cheap to buy it.

Finally, after some good programming tips from the math and computer geniuses at my forum, I made some programming shortcuts, which cut down the calculation time to under one second. Following is roughly the logic behind those shortcuts. This logic was inspired by code written by ZPP in this post on my forum.

As an example of this shortcut logic, the distribution of possible totals would be the same if the first three balls drawn were 1, 5, and 6 as if they were 2, 3, and 6. There is no need to loop through the other 18 balls twice. Likewise, the program never loops if the total of the remaining balls given the same number left and least possible ball has already been solved. This saves a HUGE amount of computer time.

I know that last statement is kind of heavy, so let me show you the critical line in my program:

a[k][level][m+t]+=a[m+1][level+1][t];

Here are the variable definitions:

Here is the full loop, after first filling in the 20th ball totals by hand.

```	for (level=19; level>=1; level--)
{
for (k=level; k<=60+level; k++) // lowest available ball
{
for (m=k; m<=60+level; m++) // level-th ball
{
for (t=0; t<=1410; t++)
{
a[k][level][m+t]+=a[m+1][level+1][t];
}
}
}
}
```

I hope that wasn't too confusing.

Lottery Sums — This same table, but based on a 6-49 lottery.