perplexus.info :: Numbers : All Combos

All Combos (Posted on 2024-05-08)

N is a positive integer such that N! expressed in decimal contains:
(A) all of the digits 0 to 9
(B) all of the 2-digit combinations 00, 01, ..., 99
(C) all of the 3-digit combinations 000, 001, ..., 999

Find the smallest value of N in conformity with the given conditions.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Computer solution

| Comment 1 of 2

(A) 23!

(B) 220!

--------- comment out 2 of 3 lines for A vs B vs C ---------

facs = ['1']

for n in range(2,3000):

facs.append(str(int(facs[-1]) * n))

ones = [str(i) for i in range(10)]

twos = [str(i).zfill(2) for i in range(100)]

threes = [str(i).zfill(3) for i in range(1000)]

for fac in facs:

hasall = True

for string in ones:

# for string in twos:

# for string in threes:

if string not in fac:

hasall = False

if hasall:

print(facs.index(fac) + 1, int(fac))

break

-----------------

(A) 23! = 25852016738884976640000

(B) 220! = 22838603359146414573972658651153337270429730715462287017736347161260276926030248

45877776549791921102945706558196074779575009550523224197049956176972302056587667

22616606097632340497755473254301355713314682574755379945084952337706589453102105

52725163342784668756149049213658078338458534285571551800849578848226429898670032

94551385992993862178352349027264696691854493614080000000000000000000000000000000

0000000000000000000000

62128978580212121661936114317148808590261490439446058993966580128655604442156838

59024112955362222273424445544950453355939184377150767020860219792816074007403794

46594587355573979614348254677477271461729420421497952998957244453425025185216460

90510766147337079088153321045830014690658657083104378674947802930809464129635250

41388412825457167629058883939837512068016244557674972675826609631879944318460982

64391584912249511846409507749644385844898122396089040219415237927938360229997411

04170598206730478327988801419447719707580587032770260472241382167869068974528198

17277558235284825982813892186386506719519080804093923119556659986509618807965048

37386106658237806834766454544761554640225181238396752519146658018987254154978329

49284870017884373853887969173895054195210387877038793117263017199588127421656427

54707208892125697143112419629755185800845271531097041005749818304261942536362066

00782863553855822173386231182984437194050059919362798742065340717500791352729885

94135008104276807071275730570792887824281379812134455043919839051807261971905412

09736291670107101441643519461838476608126443899867647560916303001587358756639073

83659305958807523762387891148352654278417679123053264839100179172760336054341706

63488679353060378152864922485784293478922501905442424653896811313188378688551677

99630138614020990114281070973253640934026930037025677676137294031141564843375932

23391873633942663616570895930329631123684798739894513552201185401889392745256289

84075910131124906927990098457384952222030680494915158905494306713000249128146058

54272128167367351318571384604258753316619971430231986641410925171110667479521606

94874712144477213636592771878995459641349959128905370667740919592395805195686627

15622339516995197120604577404941969091858807174191131959385459634122935682448462

08297539347079749458177480033542695051168049406952314617285677756045464392432146

26382271533290753193582677516647924895258839995200528311496300996784090111514435

20847748805048593126264105584462575322120821496728980444395992384529533467894429

16182291613984237237596162519521641938631015332836798098956754802946938151462822

20095743290049666682127836155738614135849933866647699567583811582206403861130725

52606021754209301385816051674733831765211059516322562977131972697158007062246481

95630691430521476369548806767986754508330491436707083525984002570236522277568137

47502245549545282913309218454997207512473763990366268979434723921919193101150685

26355149143935574534603349270921192838150035002419061711913345335619678153978609

22765621002482495812290869592162573320100822884515707122437016840803479003946315

16311068885342796717544197301939240029719290283329367752370424719363204938864732

49909837236379319761893462103607791236266728725741300014011460335716603405621805

03499388164281632881837320772485682830072778729037527152654431870799487667200132

51208715422422951096831179473698065607246722137465437858175998285424756310442610

69376717224102276193030964479807686076887111642575888623791964562261785503223992

88876568784751894616091858802371910891327447444605579586616317155487329166329947

56092365394245643965246114897472266829283156046329349000956572009303349664540403

13293187768254855391716868343682621871464327068927365344488696920106387479459934

19022318458064628123566860211015471941509792245500514002371079982958434958202268

25888261889146402911244515065676098926190089082345150976126943131636633205284887

31326390781353786477204975846297334399806887561254352384941090195466230979990283

76127235846953251583958558573057923417103371091404520424208625516575504585863215

20566538298150012220090444797043641017790380107963737604968951508669673033395588

07478579182307177073010630518274638724499724926103142267187728755778216279408861

42280516798254093383867870673393814278365925922902775153646837917032614205617526

26436705914356139115556486861766882183007307883212039587445098682824106785807809

32397853674832629758438006892770920774035398212521747280290950387664917449622088

60494631404590043227045615979894739229214501665059733555686216061934862868397922

20026909376499831905151834812207790927045177589476107928127631825962824421289170

51237044843703252629049955436239642265809776661277976317846365001067059284420675

66239803428638800314443889950313894585559514090342603820894305536320052997818565

62070898936319111471087200647024573741589742404915180844573142833220383320678586

78464850524591501501030800709032003522982991154933057167642385358295835829705565

05766649443521796381358840306980046048835884533409785634503451420539779186551571

54853420733376722390995977172103678723741566955295123057065237773951407479126567

73195989346486721824769723739353543937212182511885973507442665977813412325096076

95743085428971731884455144170956680279152219032266400002427921990675542508036352

59261596733824280677715791230492009368819100004556249659565368991089891775683723

08342488151868584570335660857239295993211755458083091779965936375775101341916926

96668445968126977821382186150320654181877765429351131438916715291472068010511296

73907207471199653794505158674137339732504632905095600187935925364861336164967473

67491221939860202942078586249681449263367663118131837216468649301868672162462607

69831234407866023425811245467603191059292608739219283585782571558907167681491682

11388319185968727780198597572226528009694932518776607224826900996018403426372936

66823561368889506531410851635560440277978960931066628445526046708313081985528615

30793487035463179745736151030245065932253955374889691651229226832390170607616000

00000000000000000000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000

Posted by Larry on 2024-05-08 07:50:26

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