Casino style Keno. 80 balls and draw 20 without replacement. We want the sums.

Pari GP code of: b=(prod(i=1, 80, 1 + x*y^i));ways=polcoeff(b,20);

was used to produce the results. We only wanted 20 draws.

The sheet below helps with the probabilities and we know Excel/Google is only accurate to 15 digits

for those that want the exact values. Below is a list of them. the value before the y is the number of ways for that sum.

The result returned from the Pari GP code (shortened)

draw=20;lowV=1;highV=80;

\\do not change below

lowSum=([1..draw]);lowSumCa=vecsum(lowSum);

highSum=([highV-draw+1..highV]);highSumCa=vecsum(highSum);

b=(prod(i=lowV, highV, 1 + x*y^i));

ways=polcoeff(b,draw);waysC=Colrev(ways);

resultCprob=Col(waysC/binomial(highV,draw));

resultCprobDec=Col(1.*resultCprob)*100;seq1C=Col([lowSumCa..highSumCa]);

resultCprobDecCUM=vector(highSumCa+1);

for(i=lowSumCa+1,highSumCa+1,resultCprobDecCUM[i]=vecsum(resultCprobDec[lowSumCa+1..i]));

resultCprobDecCUMcc=Col(resultCprobDecCUM);

\p22

xxx=matconcat([seq1C,waysC[lowSumCa+1..highSumCa+1],resultCprobDec[lowSumCa+1..highSumCa+1],\

resultCprobDecCUMcc[lowSumCa+1..highSumCa+1]])

[Sum X, ways X, percentage Sum X, percentage Sum X or less]

[ 210 1 2.828601346453457692714 E-17 2.828601346453457692714 E-17]

[ 211 1 2.828601346453457692714 E-17 5.657202692906915385428 E-17]

[ 212 2 5.657202692906915385428 E-17 1.131440538581383077086 E-16]

[ 213 3 8.485804039360373078141 E-17 1.980020942517420384900 E-16]

[ 214 5 1.414300673226728846357 E-16 3.394321615744149231257 E-16]

[ 215 7 1.980020942517420384900 E-16 5.374342558261569616156 E-16]

[ 216 11 3.111461481098803461985 E-16 8.485804039360373078141 E-16]

[ 217 15 4.242902019680186539071 E-16 1.272870605904055961721 E-15]

[ 218 22 6.222922962197606923970 E-16 1.895162902123816654118 E-15]

[ 219 30 8.485804039360373078141 E-16 2.743743306059853961932 E-15]

[ 220 42 1.188012565510452230940 E-15 3.931755871570306192872 E-15]

[ 221 56 1.584016754013936307920 E-15 5.515772625584242500792 E-15]

[ 222 77 2.178023036769162423390 E-15 7.693795662353404924182 E-15]

[ 223 101 2.856887359917992269641 E-15 1.055068302227139719382 E-14]

[ 224 135 3.818611817712167885164 E-15 1.436929483998356507899 E-14]

[ 225 176 4.978338369758085539176 E-15 1.934763320974165061816 E-14]

[ 226 231 6.534069110307487270169 E-15 2.588170232004913788833 E-14]

[ 227 297 8.400945998966769347360 E-15 3.428264831901590723569 E-14]

[ 228 385 1.089011518384581211695 E-14 4.517276350286171935264 E-14]

[ 229 490 1.386014659762194269430 E-14 5.903291010048366204694 E-14]

[ 230 627 1.773533044226317973332 E-14 7.676824054274684178025 E-14]

[ 231 791 2.237423665044685034937 E-14 9.914247719319369212962 E-14]

[ 232 1000 2.828601346453457692714 E-14 1.274284906577282690568 E-13]

[ 233 1251 3.538580284413275573585 E-14 1.628142935018610247926 E-13]

[ 234 1568 4.435246911239021662175 E-14 2.071667626142512414144 E-13]

[ 235 1946 5.504458220198428670021 E-14 2.622113448162355281146 E-13]

[ 236 2417 6.836729454378007243289 E-14 3.305786393600156005475 E-13]

[ 237 2980 8.429232012431303924287 E-14 4.148709594843286397903 E-13]

[ 238 3673 1.038945274552355010534 E-13 5.187654869395641408437 E-13]

[ 239 4498 1.272304885634765270183 E-13 6.459959755030406678620 E-13]

[ 240 5507 1.557710761491919151377 E-13 8.017670516522325829997 E-13]

[ 241 6703 1.896011482527752691426 E-13 9.913681999050078521423 E-13]

[ 242 8154 2.306441537898149402639 E-13 1.222012353694822792406 E-12]

[ 243 9871 2.792112389084208088478 E-13 1.501223592603243601254 E-12]

[ 244 11937 3.376501427261492447792 E-13 1.838873735329392846033 E-12]

[ 245 14375 4.066114435526845433276 E-13 2.245485178882077389361 E-12]

[ 246 17293 4.891500308421964388010 E-13 2.734635209724273828162 E-12]

[ 247 20722 5.861427710120855030842 E-13 3.320777980736359331246 E-12]

[ 248 24803 7.015779919608511115238 E-13 4.022355972697210442770 E-12]

[ 249 29588 8.369265663886490621202 E-13 4.859282539085859504890 E-12]

[ 250 35251 9.971102606383083712585 E-13 5.856392799724167876148 E-12]

[ 251 41869 1.184307097746598201362 E-12 7.040699897470766077511 E-12]

[ 252 49668 1.404909716756503366817 E-12 8.445609614227269444328 E-12]

[ 253 58754 1.661916435095264532777 E-12 1.010752604932253397710 E-11]

[ 254 69414 1.963445338627203122820 E-12 1.207097138794973709993 E-11]

[ 255 81801 2.313824187412392927217 E-12 1.438479557536213002714 E-11]

[ 256 96271 2.723122802244208255352 E-12 1.710791837760633828249 E-11]

[ 257 113039 3.197422676017524041267 E-12 2.030534105362386232376 E-11]

[ 258 132559 3.749565658845238982884 E-12 2.405490671246910130665 E-11]

[ 259 155112 4.387500120510887296322 E-12 2.844240683297998860297 E-11]

[ 260 181274 5.127518804770040897890 E-12 3.356992563775002950086 E-11]

[ 261 211428 5.980455254779616530551 E-12 3.955038089252964603141 E-11]

[ 262 246288 6.966505684153291882231 E-12 4.651688657668293791364 E-11]

[ 263 286364 8.100095959757979587163 E-12 5.461698253644091750080 E-11]

[ 264 332557 9.406711779725225299158 E-12 6.402369431616614279996 E-11]

[ 265 385528 1.090505019895508637357 E-11 7.492874451512122917353 E-11]

[ 266 446405 1.262701784063555781316 E-11 8.755576235575678698669 E-11]

[ 267 516054 1.459711039242692656156 E-11 1.021528727481837135482 E-10]

[ 268 595872 1.685484341513914742273 E-11 1.190077161633228609710 E-10]

[ 269 686983 1.943201038790635726114 E-11 1.384397265512292182321 E-10]

[ 270 791131 2.237794211821070437894 E-11 1.608176686694399226111 E-10]

[ 271 909741 2.573294617523915054827 E-11 1.865506148446790731593 E-10]

[ 272 1044984 2.955843149422320033563 E-11 2.161090463389022734949 E-10]

[ 273 1198689 3.390613319378948748221 E-11 2.500151795326917609772 E-10]

[ 274 1373524 3.885151835786139023927 E-11 2.888666978905531512164 E-10]

[ 275 1571812 4.446029539571702242900 E-11 3.333269932862701736454 E-10]

[ 276 1796855 5.082586472381627722441 E-11 3.841528580100864508698 E-10]

[ 277 2051569 5.803070835742173745183 E-11 4.421835663675081883217 E-10]

[ 278 2340024 6.618995037133405883935 E-11 5.083735167388422471610 E-10]

[ 279 2665885 7.540725900490076061140 E-11 5.837807757437430077724 E-10]

[ 280 3034135 8.582358346321561856482 E-11 6.696043592069586263373 E-10]

[ 281 3449359 9.756861511801352373482 E-11 7.671729743249721500721 E-10]

[ 282 3917670 1.108152663696031759901 E-10 8.779882406945753260622 E-10]

[ 283 4444748 1.257242017744631317277 E-10 1.003712442469038457790 E-9]

[ 284 5038070 1.425069158552677159793 E-10 1.146219358324306173769 E-9]

[ 285 5704686 1.613628250069418975122 E-10 1.307582183331248071281 E-9]

[ 286 6453684 1.825489925198513665614 E-10 1.490131175851099437843 E-9]

[ 287 7293767 2.063115915691779675501 E-10 1.696442767420277405393 E-9]

[ 288 8236001 2.329636351799202401065 E-10 1.929406402600197645499 E-9]

[ 289 9291060 2.628070482597986263047 E-10 2.192213450859996271804 E-9]

[ 290 10472375 2.962217402556552900473 E-10 2.488435191115651561851 E-9]

[ 291 11793028 3.335777487956332726699 E-10 2.822012939911284834521 E-9]

[ 292 13269250 3.753341841642754348904 E-10 3.197347124075560269412 E-9]

[ 293 14917024 4.219431417147854328520 E-10 3.619290265790345702264 E-9]

[ 294 16755957 4.739592253131623960043 E-10 4.093249491103508098268 E-9]

[ 295 18805464 5.319316079108202631585 E-10 4.625181099014328361427 E-9]

[ 296 21089176 5.965287162919394509020 E-10 5.221709815306267812329 E-9]

[ 297 23630664 6.684172800798925037473 E-10 5.890127095386160316076 E-9]

[ 298 26458297 7.483997451906548031076 E-10 6.638526840576815119183 E-9]

[ 299 29600587 8.372826024401271588399 E-10 7.475809443016942278023 E-9]

[ 300 33091578 9.360288208706961860814 E-10 8.411838263887638464105 E-9]

[ 301 36965620 1.045610025044868647549 E-9 9.457448288932507111654 E-9]

[ 302 41263462 1.167178841725310862719 E-9 1.062462713065781797437 E-8]

[ 303 46026431 1.301904246990471652051 E-9 1.192653137764828962642 E-8]

[ 304 51303148 1.451161535101010151210 E-9 1.337769291274929977763 E-8]

[ 305 57143201 1.616353352892605700797 E-9 1.499404626564190547843 E-8]

[ 306 63604540 1.799118874845528079545 E-9 1.679316514048743355798 E-8]

[ 307 70746444 2.001134867551941432639 E-9 1.879430000803937499062 E-8]

[ 308 78637842 2.224351057633942663933 E-9 2.101865106567331765455 E-8]

[ 309 87349501 2.470769161406376491832 E-9 2.348942022707969414638 E-8]

[ 310 96963169 2.742701503897941688930 E-9 2.623212173097763583531 E-8]

[ 311 107563103 3.042531379745119545075 E-9 2.927465311072275538039 E-8]

[ 312 119246146 3.372998091349855982302 E-9 3.264765120207261136269 E-8]

[ 313 132112328 3.736931088639008394339 E-9 3.638458229071161975703 E-8]

[ 314 146276124 4.137568413003929376882 E-9 4.052215070371554913391 E-8]

[ 315 161856093 4.578263625914960684834 E-9 4.510041432963050981874 E-8]

[ 316 178987225 5.062835056329679840737 E-9 5.016324938596018965948 E-8]

[ 317 197809729 5.595248657909935773087 E-9 5.575849804387012543257 E-8]

[ 318 218482569 6.180000888500104756369 E-9 6.193849893237023018894 E-8]

[ 319 241171101 6.821769010142628370087 E-9 6.876026794251285855902 E-8]

[ 320 266062168 7.525838066451260648197 E-9 7.628610600896411920722 E-8]

[ 321 293350487 8.297715825109775369915 E-9 8.458382183407389457714 E-8]

[ 322 323255164 9.143599921384332848253 E-9 9.372742175545822742539 E-8]

[ 323 356005000 1.006996222344163205895 E-8 1.037973839788998594843 E-7]

[ 324 391856581 1.108406052633248407095 E-8 1.148814445052323435553 E-7]

[ 325 431078138 1.219348201573449445837 E-8 1.270749265209668380137 E-7]

[ 326 473969587 1.340671011966189257338 E-8 1.404816366406287305870 E-7]

[ 327 520844871 1.473262503403977479465 E-8 1.552142616746685053817 E-7]

[ 328 572053851 1.618112293182485665383 E-8 1.713953846064933620355 E-7]

[ 329 627963266 1.776257739730910809709 E-8 1.891579620038024701326 E-7]

[ 330 688980643 1.948851574470169050699 E-8 2.086464777485041606396 E-7]

[ 331 755533468 2.137102984875450391167 E-8 2.300175075972586645513 E-7]

[ 332 828095629 2.342352411181622967273 E-8 2.534410317090748942240 E-7]

[ 333 907164667 2.566007198531202578809 E-8 2.791011036943869200121 E-7]

[ 334 993290571 2.809623046550123816520 E-8 3.071973341598881581773 E-7]

[ 335 1087051347 3.074834903788244857672 E-8 3.379456831977706067540 E-7]

[ 336 1189084411 3.363445766001416679454 E-8 3.715801408577847735485 E-7]

[ 337 1300059954 3.677351336354620271130 E-8 4.083536542213309762599 E-7]

[ 338 1420715489 4.018637745112682561645 E-8 4.485400316724578018763 E-7]

[ 339 1551826889 4.389499627688080434577 E-8 4.924350279493386062221 E-7]

[ 340 1694245990 4.792346488537371417515 E-8 5.403584928347123203972 E-7]

[ 341 1848869496 5.229714745802325711785 E-8 5.926556402927355775151 E-7]

[ 342 2016679881 5.704383426762198831781 E-8 6.496994745603575658329 E-7]

[ 343 2198711618 6.219278643137660525241 E-8 7.118922609917341710853 E-7]

[ 344 2396096082 6.777600603777054572864 E-8 7.796682670295047168139 E-7]

[ 345 2610024936 7.382720048246699741404 E-8 8.534954675119717142280 E-7]

[ 346 2841798846 8.038316042145482263864 E-8 9.338786279334265368666 E-7]

[ 347 3092788314 8.748265189275919296919 E-8 1.021361279826185729836 E-6]

[ 348 3364486581 9.516791273141190348187 E-8 1.116529192557597633318 E-6]

[ 349 3658467191 1.034834522241839917730 E-7 1.220012644781781625091 E-6]

[ 350 3976441876 1.124776884454751325430 E-7 1.332490333227256757634 E-6]

[ 351 4320214692 1.222016509475921001826 E-7 1.454691984174848857816 E-6]

[ 352 4691744778 1.327107559626677894983 E-7 1.587402740137516647315 E-6]

[ 353 5093097335 1.440634197939951707630 E-7 1.731466159931511818078 E-6]

[ 354 5526511602 1.563229815860785549180 E-7 1.887789141517590372996 E-6]

[ 355 5994347965 1.695562072490954408753 E-7 2.057345348766685813871 E-6]

[ 356 6499162145 1.838353879416634224084 E-7 2.241180736708349236279 E-6]

[ 357 7043648186 1.992367274286405481071 E-7 2.440417464136989784386 E-6]

[ 358 7630718671 2.158426110719813924830 E-7 2.656260075208971176869 E-6]

[ 359 8263443950 2.337398868331267892744 E-7 2.889999962042097966144 E-6]

[ 360 8945138936 2.530223203858284991892 E-7 3.143022282427926465333 E-6]

[ 361 9679297643 2.737887434571357945428 E-7 3.416811025885062259876 E-6]

[ 362 10469687646 2.961457257242273191939 E-7 3.712956751609289579070 E-6]

[ 363 11320279729 3.202055848367918316079 E-7 4.033162336446081410678 E-6]

[ 364 12235349889 3.460892717035456412421 E-7 4.379251608149627051920 E-6]

[ 365 13219404469 3.739242528032625592356 E-7 4.753175860952889611155 E-6]

[ 366 14277290236 4.038476238525640474462 E-7 5.157023484805453658602 E-6]

[ 367 15414114032 4.360038370530233565618 E-7 5.593027321858477015163 E-6]

[ 368 16635362300 4.705480820052108880602 E-7 6.063575403863687903224 E-6]

[ 369 17946814901 5.076438479351957802257 E-7 6.571219251798883683449 E-6]

[ 370 19354673930 5.474665673856563556353 E-7 7.118685819184540039085 E-6]

[ 371 20865472184 5.902010271404956853794 E-7 7.708886846325035724464 E-6]

[ 372 22486212019 6.360452959358132339401 E-7 8.344932142260848958404 E-6]

[ 373 24224267278 6.852079503939923653556 E-7 9.030140092654841323760 E-6]

[ 374 26087533643 7.379123278823967629235 E-7 9.768052420537238086683 E-6]

[ 375 28084323773 7.943935603854265111753 E-7 1.056244598092266459786 E-5]

[ 376 30223529082 8.549031505592043613519 E-7 1.141734913148186895921 E-5]

[ 377 32514508546 9.197058265248805694099 E-7 1.233705495800674952862 E-5]

[ 378 34967262684 9.890844630975414692120 E-7 1.332613942110429099783 E-5]

[ 379 37592314751 1.063336721209807791568 E-6 1.438947614231409878940 E-5]

[ 380 40400898786 1.142780367040094643031 E-6 1.553225650935419343243 E-5]

[ 381 43404832757 1.227749683790373462366 E-6 1.676000619314456689480 E-5]

[ 382 46616720518 1.318601184244593277560 E-6 1.807860737738916017236 E-5]

[ 383 50049817550 1.415709813116798970870 E-6 1.949431719050595914323 E-5]

[ 384 53718247833 1.519475081495443359370 E-6 2.101379227200140250260 E-5]

[ 385 57636860670 1.630317016965123396747 E-6 2.264410928896652589934 E-5]

[ 386 61821464449 1.748682775801659669832 E-6 2.439279206476818556918 E-5]

[ 387 66288674187 1.875042330599627649519 E-6 2.626783439536781321869 E-5]

[ 388 71056162189 2.009895560416206697533 E-6 2.827772995578401991623 E-5]

[ 389 76142496873 2.153767691772959920075 E-6 3.043149764755697983630 E-5]

[ 390 81567411419 2.307216897665065401567 E-6 3.273871454522204523787 E-5]

[ 391 87351632249 2.470829445944286767684 E-6 3.520954399116633200555 E-5]

[ 392 93517168095 2.645227875900313351435 E-6 3.785477186706664535699 E-5]

[ 393 100087127594 2.831065838750474194731 E-6 4.068583770581711955172 E-5]

[ 394 107086028628 3.029036847635143167515 E-6 4.371487455345226271923 E-5]

[ 395 114539605867 3.239868833776445623828 E-6 4.695474338722870834306 E-5]

[ 396 122475141756 3.464333508780996987715 E-6 5.041907689600970533078 E-5]

[ 397 130921262045 3.703240580998729659841 E-6 5.412231747700843499062 E-5]

[ 398 139908291043 3.957447804242320347259 E-6 5.807976528125075533788 E-5]

[ 399 149468034620 4.227854839778840286333 E-6 6.230762012102959562421 E-5]

[ 400 159634159836 4.515413994520760770535 E-6 6.682303411555035639475 E-5]

[ 401 170441966142 4.821123749214358478404 E-6 7.164415786476471487315 E-5]

[ 402 181928790813 5.146040226523012440183 E-6 7.679019809128772731333 E-5]

[ 403 194133766301 5.491270327510894908907 E-6 8.228146841879862222224 E-5]

[ 404 207098254570 5.857984017248629479024 E-6 8.813945243604725170126 E-5]

[ 405 220865589927 6.247407050527494425502 E-6 9.438685948657474612676 E-5]

[ 406 235481542190 6.660834073035707045382 E-6 0.0001010476935596104531721]

[ 407 250994045973 7.099620963910288606298 E-6 0.0001081473145235207417784]

[ 408 267453694512 7.565198804105949483732 E-6 0.0001157125133276266912622]

[ 409 284913452879 8.059065764362431720996 E-6 0.0001237715790919891229832]

[ 410 303429185857 8.582802036683866620427 E-6 0.0001323543811286729896036]

[ 411 323059354425 9.138061249109398055172 E-6 0.0001414924423777823876588]

[ 412 343865577844 9.726586364885346695568 E-6 0.0001512190287426677343543]

[ 413 365912314318 1.035020064963795625814 E-5 0.0001615692293923056906125]

[ 414 389267459099 1.101082458937947671106 E-5 0.0001725800539816851673235]

[ 415 414002006749 1.171046633724654878913 E-5 0.0001842905203189317161127]

[ 416 440190689221 1.245123976226796145754 E-5 0.0001967417600811996775702]

[ 417 467911618783 1.323535434910810804965 E-5 0.0002099771144303077856199]

[ 418 497246966079 1.406513437770956204281 E-5 0.0002240422488080173476627]

[ 419 528282584833 1.494300830766536787237 E-5 0.0002389852571156827155350]

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