The Math Behind Solitaire Mah Jongg

Formulas and Calculations for Tile Placements in Mah Jongg

Last Updated - 17 Feb 03

Introduction

Mahjongg is an ancient Chinese game most commonly played with 144 square tiles (36 groups of 4) aligned in preset arranged starting positions. In Asia it is most often played as a competitive game between three or more players but solitaire versions have afforded single players hours of enjoyment and that is the basis for the information on this page.

The object of the multi-player games is to form scoring groups of 3 or more tiles and outscore their opponents with scoring combinations. The object of the single-player game is to match pairs of tiles from a pre-arranged starting grid and remove them from the table until none are left.

As in most Oriental games, the number of ways the game can start and the numbers of combinations possible seem almost limitless, and that indeed is the case here. If one would disregard groups of tiles, the total number of starting positions can be expressed as 144! (factorial 144).

The tables in this page and the data within were derived in two different ways. Unique placement data is when a particular group has been selected for placement, total placement data is random group selection. Regardless, the total placement data has been calculated using the formulas derived and checked by the unique placement data. The cumulative unique divisor then would be factorial 36 calculated in reverse. The data may not be totally correct due to typos, but it's as good as I can make it. If anyone should discover any errors, I urge you to contact me so I can check the spreadsheets used in the calculations and make the necessary corrections.

Links

Yahoo Solitaire. A variety of starting shapes. No undo.

Tables

Groups Present 1
Unique Divisor 36
Formula for Cumulative Unique Divisor (36!/35!)
Cumulative Unique Divisor 36
Formula for Unique Placements ((144!/140!)/(4!))/(36)
Unique Placement Possibilities 477191
Formula For Cumulative Unique Placements ((144!/140!)/(4!)1)/(36!/35!)
Cumulative Unique Placements 477191
Formula for Total Placements (144!/140!)/(4!)
Total Placement Possibilities 17178876
Formula for Cumulative Total Placements (144!/140!)/(4!)1
Cumulative Total Placements 17178876

Groups Present 2
Unique Divisor 35
Formula for Cumulative Unique Divisor (36!/34!)
Cumulative Unique Divisor 1260
Formula for Unique Placements ((140!/136!)/(4!))/(35)
Unique Placement Possibilities 437989
Formula For Cumulative Unique Placements ((144!/136!)/(4!)2)/(36!/34!)
Cumulative Unique Placements 209004408899
Formula for Total Placements (140!/136!)/(4!)
Total Placement Possibilities 15329615
Formula for Cumulative Total Placements (144!/136!)/(4!)2
Cumulative Total Placements 263345555212740

Groups Present 3
Unique Divisor 34
Formula for Cumulative Unique Divisor (36!/33!)
Cumulative Unique Divisor 42840
Formula for Unique Placements ((136!/132!)/(4!))/(34)
Unique Placement Possibilities 400995
Formula For Cumulative Unique Placements ((144!/132!)/(4!)3)/(36!/33!)
Cumulative Unique Placements 83809722946454505
Formula for Total Placements (136!/132!)/(4!)
Total Placement Possibilities 13633830
Formula for Cumulative Total Placements (144!/132!)/(4!)3
Cumulative Total Placements 3590408531026110994200

Groups Present 4
Unique Divisor 33
Formula for Cumulative Unique Divisor (36!/32!)
Cumulative Unique Divisor 1413720
Formula for Unique Placements ((132!/128!)/(4!))/(33)
Unique Placement Possibilities 366145
Formula For Cumulative Unique Placements ((144!/128!)/(4!)4)/(36!/32!)
Cumulative Unique Placements 30686511008229584733225
Formula for Total Placements (132!/128!)/(4!)
Total Placement Possibilities 12082785
Formula for Cumulative Total Placements (144!/128!)/(4!)4
Cumulative Total Placements 43382134342554328529054847000

Groups Present 5
Unique Divisor 32
Formula for Cumulative Unique Divisor (36!/31!)
Cumulative Unique Divisor 45239040
Formula for Unique Placements ((128!/124!)/(4!))/(32)
Unique Placement Possibilities 333375
Formula For Cumulative Unique Placements ((144!/124!)/(4!)5)/(36!/31!)
Cumulative Unique Placements 10230115607368537810438884375
Formula for Total Placements (128!/124!)/(4!)
Total Placement Possibilities 10668000
Formula for Cumulative Total Placements (144!/124!)/(4!)5
Cumulative Total Placements 462800609166369576747597107796000000

Groups Present 6
Unique Divisor 31
Formula for Cumulative Unique Divisor (36!/30!)
Cumulative Unique Divisor 1402410240
Formula for Unique Placements ((124!/120!)/(4!))/(31)
Unique Placement Possibilities 302621
Formula For Cumulative Unique Placements ((144!/120!)/(4!)6)/(36!/30!)
Cumulative Unique Placements 3095847815217474280732825628446875
Formula for Total Placements (124!/120!)/(4!)
Total Placement Possibilities 9381251
Formula for Cumulative Total Placements (144!/120!)/(4!)6
Cumulative Total Placements 4341648677542613758236349365468332796000000

Groups Present 7
Unique Divisor 30
Formula for Cumulative Unique Divisor (36!/29!)
Cumulative Unique Divisor 42072307200
Formula for Unique Placements ((120!/116!)/(4!))/(30)
Unique Placement Possibilities 273819
Formula For Cumulative Unique Placements ((144!/116!)/(4!)7)/(36!/29!)
Cumulative Unique Placements 847701952915033590075981580755694865625
Formula for Total Placements (120!/116!)/(4!)
Total Placement Possibilities 8214570
Formula for Cumulative Total Placements (144!/116!)/(4!)7
Cumulative Total Placements 35664776977081228699995568407095202536037720000000

Groups Present 8
Unique Divisor 29
Formula for Cumulative Unique Divisor (36!/28!)
Cumulative Unique Divisor 1220096908800
Formula for Unique Placements ((116!/112!)/(4!))/(29)
Unique Placement Possibilities 246905
Formula For Cumulative Unique Placements ((144!/112!)/(4!)8)/(36!/28!)
Cumulative Unique Placements 209301850684486368557710232196484840797140625
Formula for Total Placements (116!/112!)/(4!)
Total Placement Possibilities 7160245
Formula for Cumulative Total Placements (144!/112!)/(4!)8
Cumulative Total Placements 255368541026260982392999768709061388482651404441400000000

Groups Present 9
Unique Divisor 28
Formula for Cumulative Unique Divisor (36!/27!)
Cumulative Unique Divisor 34162713446400
Formula for Unique Placements ((112!/108!)/(4!))/(28)
Unique Placement Possibilities 221815
Formula For Cumulative Unique Placements ((144!/108!)/(4!)9)/(36!/27!)
Cumulative Unique Placements 46426290009579343841628495154663284961417747734375
Formula for Total Placements (112!/108!)/(4!)
Total Placement Possibilities 6210820
Formula for Cumulative Total Placements (144!/108!)/(4!)9
Cumulative Total Placements 1586048041976722234666090823493612652825820995732735948000000000

Groups Present 10
Unique Divisor 27
Formula for Cumulative Unique Divisor (36!/26!)
Cumulative Unique Divisor 922393263052800
Formula for Unique Placements ((108!/104!)/(4!))/(27)
Unique Placement Possibilities 198485
Formula For Cumulative Unique Placements ((144!/104!)/(4!)10)/(36!/26!)
Cumulative Unique Placements 9214922172551356062405631860773342115567001659057421875
Formula for Total Placements (108!/104!)/(4!)
Total Placement Possibilities 5359095
Formula for Cumulative Total Placements (144!/104!)/(4!)10
Cumulative Total Placements 8499782131517242244187874001730502099642002219126326555247060000000000

Groups Present 11
Unique Divisor 26
Formula for Cumulative Unique Divisor (36!/25!)
Cumulative Unique Divisor 23982224839372800
Formula for Unique Placements ((104!/100!)/(4!))/(26)
Unique Placement Possibilities 176851
Formula For Cumulative Unique Placements ((144!/100!)/(4!)11)/(36!/25!)
Cumulative Unique Placements 1629668201137879870992498400209626326480139810405964416015625
Formula for Total Placements (104!/100!)/(4!)
Total Placement Possibilities 4598126
Formula for Cumulative Total Placements (144!/100!)/(4!)11
Cumulative Total Placements 39083069213264851011298612332081066697418481095822459418171943009560000000000

Groups Present 12
Unique Divisor 25
Formula for Cumulative Unique Divisor (36!/24!)
Cumulative Unique Divisor 599555620984320000
Formula for Unique Placements ((100!/96!)/(4!))/(25)
Unique Placement Possibilities 156849
Formula For Cumulative Unique Placements ((144!/96!)/(4!)12)/(36!/24!)
Cumulative Unique Placements 255611827680275319885302381574479679682083449122365065632934765625
Formula for Total Placements (100!/96!)/(4!)
Total Placement Possibilities 3921225
Formula for Cumulative Total Placements (144!/96!)/(4!)12
Cumulative Total Placements 153253508075784465406779401141864580760584783534966423432021277227661911000000000000

Groups Present 13
Unique Divisor 24
Formula for Cumulative Unique Divisor (36!/23!)
Cumulative Unique Divisor 14389334903623680000
Formula for Unique Placements ((96!/92!)/(4!))/(24)
Unique Placement Possibilities 138415
Formula For Cumulative Unique Placements ((144!/92!)/(4!)13)/(36!/23!)
Cumulative Unique Placements 35380511128365308401924129145631604863195580610272160559582665583984375
Formula for Total Placements (96!/92!)/(4!)
Total Placement Possibilities 3321960
Formula for Cumulative Total Placements (144!/92!)/(4!)13
Cumulative Total Placements 509102023687432962702704899417228462703432227511817059984237402099203761865560000000000000

Groups Present 14
Unique Divisor 23
Formula for Cumulative Unique Divisor (36!/22!)
Cumulative Unique Divisor 330954702783344640000
Formula for Unique Placements ((92!/88!)/(4!))/(23)
Unique Placement Possibilities 121485
Formula For Cumulative Unique Placements ((144!/88!)/(4!)14)/(36!/22!)
Cumulative Unique Placements 4298201394429459491207752829257055516805315110438913425580900128470341796875
Formula for Total Placements (92!/88!)/(4!)
Total Placement Possibilities 2794155
Formula for Cumulative Total Placements (144!/88!)/(4!)14
Cumulative Total Placements 1422509964996359249900576408231145995205108675663281197240256858262500687235463801000000000000000

Groups Present 15
Unique Divisor 22
Formula for Cumulative Unique Divisor (36!/21!)
Cumulative Unique Divisor 7281003461233582080000
Formula for Unique Placements ((88!/84!)/(4!))/(22)
Unique Placement Possibilities 105995
Formula For Cumulative Unique Placements ((144!/84!)/(4!)15)/(36!/21!)
Cumulative Unique Placements 455587856802550558770565761137101599503779375130972628544447509117213878759765625
Formula for Total Placements (88!/84!)/(4!)
Total Placement Possibilities 2331890
Formula for Cumulative Total Placements (144!/188)/(4!)15
Cumulative Total Placements 3317136762275360171250655120590127034758840869692448791032582565213742727557505684779402000000000000000

Groups Present 16
Unique Divisor 21
Formula for Cumulative Unique Divisor (36!/20!)
Cumulative Unique Divisor 152901072685905223680000
Formula for Unique Placements ((84!/80!)/(4!))/(21)
Unique Placement Possibilities 91881
Formula For Cumulative Unique Placements ((144!/80!)/(4!)16)/(36!/20!)
Cumulative Unique Placements 41859867870875147890398352699038032064006752766408896083292381585198728394326025390625
Formula for Total Placements (84!/80!)/(4!)
Total Placement Possibilities 1929501
Formula for Cumulative Total Placements (144!/80!)/(4!)16
Cumulative Total Placements 6400418699947069725788310305833770703694218216912449634746159092162481806564934776287540938402000000000000000

Groups Present 17
Unique Divisor 20
Formula for Cumulative Unique Divisor (36!/19!)
Cumulative Unique Divisor 3058021453718104473600000
Formula for Unique Placements ((80!/76!)/(4!))/(20)
Unique Placement Possibilities 79079
Formula For Cumulative Unique Placements ((144!/76!)/(4!)17)/(36!/19!)
Cumulative Unique Placements 3310236491360935820024811333087228537589590002014849093370678243375930242694907761865234375
Formula for Total Placements (80!/76!)/(4!)
Total Placement Possibilities 1581580
Formula for Cumulative Total Placements (144!/76!)/(4!)17
Cumulative Total Placements 10122774207462286536912275813500575069548701647504392093321830296982337975626969543480848997357835160000000000000000

Groups Present 18
Unique Divisor 19
Formula for Cumulative Unique Divisor (36!/18!)
Cumulative Unique Divisor 58102407620643984998400000
Formula for Unique Placements ((76!/72!)/(4!))/(19)
Unique Placement Possibilities 67525
Formula For Cumulative Unique Placements ((144!/72!)/(4!)18)/(36!/18!)
Cumulative Unique Placements 223523719079147191247175385266715107000737064886052685029855048383959689637973646619949951171875
Formula for Total Placements (76!/72!)/(4!)
Total Placement Possibilities 1282975
Formula for Cumulative Total Placements (144!/72!)/(4!)18
Cumulative Total Placements 12987266238818927069695027061825900299854245496206947445929575225270915064280011250047342242385168564401000000000000000000

Groups Present 19
Unique Divisor 18
Formula for Cumulative Unique Divisor (36!/17!)
Cumulative Unique Divisor 1045843337171591729971200000
Formula for Unique Placements ((72!/68!)/(4!))/(18)
Unique Placement Possibilities 57155
Formula For Cumulative Unique Placements ((144!/68!)/(4!)19)/(36!/17!)
Cumulative Unique Placements 12775498163968657715732309144919101940627126943562341212881365290385216061258383772563239459228515625
Formula for Total Placements (72!/68!)/(4!)
Total Placement Possibilities 1028790
Formula for Cumulative Total Placements (144!/68!)/(4!)19
Cumulative Total Placements 13361169633834523980031946890935867969487048224042745462897887696006464708980632773936205225543437567370104790000000000000000000

Groups Present 20
Unique Divisor 17
Formula for Cumulative Unique Divisor (36!/16!)
Cumulative Unique Divisor 17779336731917059409510400000
Formula for Unique Placements ((68!/64!)/(4!))/(17)
Unique Placement Possibilities 47905
Formula For Cumulative Unique Placements ((144!/64!)/(4!)20)/(36!/16!)
Cumulative Unique Placements 612010239544918547872156269587349578465742516231353955803081804235903775414582874624641988294342041015625
Formula for Total Placements (68!/64!)/(4!)
Total Placement Possibilities 814385
Formula for Cumulative Total Placements (144!/64!)/(4!)20
Cumulative Total Placements 10881136132250328811477991314774806836330710582322051263802096271312224762023192621602036492604912403302702789404150000000000000000000

Groups Present 21
Unique Divisor 16
Formula for Cumulative Unique Divisor (36!/15!)
Cumulative Unique Divisor 284469387710672950552166400000
Formula for Unique Placements ((64!/60!)/(4!))/(16)
Unique Placement Possibilities 39711
Formula For Cumulative Unique Placements ((144!/60!)/(4!)21)/(36!/15!)
Cumulative Unique Placements 24303538622568260454551197621583239110453101062063296938896181528011974825488500534219157917734616790771484375
Formula for Total Placements (64!/60!)/(4!)
Total Placement Possibilities 635376
Formula for Cumulative Total Placements (144!/60!)/(4!)21
Cumulative Total Placements 6913612751164684918921640209616357668440641566953455643789520720481276120395248035143015538524881352440858087520451210400000000000000000000

Groups Present 22
Unique Divisor 15
Formula for Cumulative Unique Divisor (36!/14!)
Cumulative Unique Divisor 4267040815660094258282496000000
Formula for Unique Placements ((60!/56!)/(4!))/(15)
Unique Placement Possibilities 32509
Formula For Cumulative Unique Placements ((144!/56!)/(4!)22)/(36!/14!)
Cumulative Unique Placements 790083737081071579117004883480149520241719862426615720186575965294141289601805663866930604747634657251190185546875
Formula for Total Placements (60!/56!)/(4!)
Total Placement Possibilities 487635
Formula for Cumulative Total Placements (144!/56!)/(4!)22
Cumulative Total Placements 3371919553914191130438354023616272571649964476201348342859302936531887080968936775616964382128580518297497833508035225983404000000000000000000000

Groups Present 23
Unique Divisor 14
Formula for Cumulative Unique Divisor (36!/13!)
Cumulative Unique Divisor 59738571419241319615954944000000
Formula for Unique Placements ((56!/52!)/(4!))/(14)
Unique Placement Possibilities 26235
Formula For Cumulative Unique Placements ((144!/52!)/(4!)22)/(36!/13!)
Cumulative Unique Placements 20727846842321912878134623118099099163541520590762263419094820449491796732703371591548294415554195232984974517822265625
Formula for Total Placements (56!/52!)/(4!)
Total Placement Possibilities 367290
Formula for Cumulative Total Placements (144!/52!)/(4!)23
Cumulative Total Placements 1238251958957143260298703049334020752841215452463993232848793375558796805969080788316354847912006338565487979269166258151444455160000000000000000000000

Groups Present 24
Unique Divisor 13
Formula for Cumulative Unique Divisor (36!/12!)
Cumulative Unique Divisor 776601428450137155007414272000000
Formula for Unique Placements ((52!/48!)/(4!))/(13)
Unique Placement Possibilities 20825
Formula For Cumulative Unique Placements ((144!/48!)/(4!)24)/(36!/12!)
Cumulative Unique Placements 431657410491353835687153526434413740080752166302624135702649635860666666958547713394006350953916115726912094333648681640625
Formula for Total Placements (52!/48!)/(4!)
Total Placement Possibilities 270725
Formula for Cumulative Total Placements (144!/48!)/(4!)24
Cumulative Total Placements 335225761588672609144366383030952768312965125868314567962989586598155265295797396416945166200977916008141733187645035238049800123191000000000000000000000000

Groups Present 25
Unique Divisor 12
Formula for Cumulative Unique Divisor (36!/11!)
Cumulative Unique Divisor 9319217141401645860088971264000000
Formula for Unique Placements ((48!/44!)/(4!))/(12)
Unique Placement Possibilities 16215
Formula For Cumulative Unique Placements ((144!/44!)/(4!)25)/(36!/11!)
Cumulative Unique Placements 6999324911117302445667194431134018795409396376597050360418463845480710004732851172683812980717749816511879609620113372802734375
Formula for Total Placements (48!/44!)/(4!)
Total Placement Possibilities 194580
Formula for Cumulative Total Placements (144!/44!)/(4!)25
Cumulative Total Placements 65228228689923916287310810810162789658336754191456648634238513760269051521291670594809190439386282896864218443651970956619730107970504780000000000000000000000000

Groups Present 26
Unique Divisor 11
Formula for Cumulative Unique Divisor (36!/10!)
Cumulative Unique Divisor 102511388555418104460978683904000000
Formula for Unique Placements ((44!/40!)/(4!))/(11)
Unique Placement Possibilities 12341
Formula For Cumulative Unique Placements ((44!/!)/(40!)26)/(36!/10!)
Cumulative Unique Placements 86378668728098629481978846474624925954147360683584198497924262317077442168408116322090935995037750485573106262321819133758544921875
Formula for Total Placements (44!/40!)/(4!)
Total Placement Possibilities 135751
Formula for Cumulative Total Placements (144!/40!)/(4!)26
Cumulative Total Placements 8854797272885861559918729878290408858908872718244431508746512481470284013066865623786302411337127289532214517944198709332084981887103994389780000000000000000000000000

Groups Present 27
Unique Divisor 10
Formula for Cumulative Unique Divisor (36!/9!)
Cumulative Unique Divisor 1025113885554181044609786839040000000
Formula for Unique Placements ((40!/36!)/(4!))/(10)
Unique Placement Possibilities 9139
Formula For Cumulative Unique Placements ((40!/36!)/(4!)27)/(36!/9!)
Cumulative Unique Placements 789414653506093374835804677931597198294952729287275990072529833315770743977081775067589064058650001687652618131359105063419342041015625
Formula for Total Placements (40!/36!)/(4!)
Total Placement Possibilities 91390
Formula for Cumulative Total Placements (144!/36!)/(4!)27
Cumulative Total Placements 809239922769038887960972723576960465615681877720358595584343775681569255954180849357830177372100062990349084794920320045859246494662434047281994200000000000000000000000000

Groups Present 28
Unique Divisor 9
Formula for Cumulative Unique Divisor (36!/8!)
Cumulative Unique Divisor 9226024969987629401488081551360000000
Formula for Unique Placements ((36!/32!)/(4!))/(9)
Unique Placement Possibilities 6545
Formula For Cumulative Unique Placements ((144!/32!)/(4!)28)/(36!/8!)
Cumulative Unique Placements 5166718907197381138300341617062303662840465613185221355024707759051719519330000217817370424263864261045686385669745342640079593658447265625
Formula for Total Placements (36!/32!)/(4!)
Total Placement Possibilities 58905
Formula for Cumulative Total Placements (144!/32!)/(4!)28
Cumulative Total Placements 47668277650710235695341098282300856227091741007117723072895770106522837021981022931422986598103554210446512839844781452301338914768090677555145868351000000000000000000000000000

Groups Present 29
Unique Divisor 8
Formula for Cumulative Unique Divisor (36!/7!)
Cumulative Unique Divisor 73808199759901035211904652410880000000
Formula for Unique Placements ((32!/28!)/(4!))/(8)
Unique Placement Possibilities 4495
Formula For Cumulative Unique Placements ((144!/28!)/(4!)29)/(36!/7!)
Cumulative Unique Placements 23224401487852228216660035568695054964467892931267569990836061736937479239388350979089080057066069853400360303585505315167157773494720458984375
Formula for Total Placements (32!/28!)/(4!)
Total Placement Possibilities 35960
Formula for Cumulative Total Placements (144!/28!)/(4!)29
Cumulative Total Placements 1714151264319540075604465894231538789926219006615953321701331893030561219310437584613970598067803809407656601720818341024756147375060540764883045425901960000000000000000000000000000

Groups Present 30
Unique Divisor 7
Formula for Cumulative Unique Divisor (36!/6!)
Cumulative Unique Divisor 516657398319307246483332566876160000000
Formula for Unique Placements ((28!/24!)/(4!))/(7)
Unique Placement Possibilities 2925
Formula For Cumulative Unique Placements ((144!/24!)/(4!)30)/(36!/6!)
Cumulative Unique Placements 67931374351967767533730604038433035771068586823957642223195479527542126775210926613835559166918254321196053887987603046863936487472057342529296875
Formula for Total Placements (28!/24!)/(4!)
Total Placement Possibilities 20475
Formula for Cumulative Total Placements (144!/24!)/(4!)30
Cumulative Total Placements 35097247136942583048001439184390756123739334160461644261834770509800740965381209544971047995438282997621738920233755532481882117504364572160980355095342631000000000000000000000000000000

Groups Present 31
Unique Divisor 6
Formula for Cumulative Unique Divisor (36!/5!)
Cumulative Unique Divisor 3099944389915843478899995401256960000000
Formula for Unique Placements ((24!/20!)/(4!))/(6)
Unique Placement Possibilities 1771
Formula For Cumulative Unique Placements ((144!/20!)/(4!)31)/(36!/5!)
Cumulative Unique Placements 120306463977334916302236899752064906350562467265228984377279194243277106518898551033102775284612228402838211435626044995996031519313013553619384765625
Formula for Total Placements (24!/20!)/(4!)
Total Placement Possibilities 10626
Formula for Cumulative Total Placements (144!/20!)/(4!)31
Cumulative Total Placements 372943348077151887468036292773336180946454164789065431926256271437142673498140732624862355999527195132728916546403886288152479380601377943782577253243110797006000000000000000000000000000000

Groups Present 32
Unique Divisor 5
Formula for Cumulative Unique Divisor (36!/4!)
Cumulative Unique Divisor 15499721949579217394499977006284800000000
Formula for Unique Placements ((20!/16!)/(4!))/(5)
Unique Placement Possibilities 969
Formula For Cumulative Unique Placements ((144!/16!)/(4!)32)/(36!/4!)
Cumulative Unique Placements 116576963594037533896867555859750894253695030780006885861583539221735516216812695951076589250789249322350226881121637601120154542214310133457183837890625
Formula for Total Placements (20!/16!)/(4!)
Total Placement Possibilities 4845
Formula for Cumulative Total Placements (144!/16!)/(4!)32
Cumulative Total Placements 1806910521433800894782766653486813796685570428403022017682711635112956253098491849567458114817709260418071600667326829066098762599013676137626586791962871811494070000000000000000000000000000000

Groups Present 33
Unique Divisor 4
Formula for Cumulative Unique Divisor (36!/3!)
Cumulative Unique Divisor 61998887798316869577999908025139200000000
Formula for Unique Placements ((16!/12!)/(4!))/(4)
Unique Placement Possibilities 455
Formula For Cumulative Unique Placements ((144!/12!)/(4!)33)/(36!/3!)
Cumulative Unique Placements 53042518435287077923074737916186656885431239004903133067020510345889659878649776657739848109109108441669353230910345108509670316707511110723018646240234375
Formula for Total Placements (16!/12!)/(4!)
Total Placement Possibilities 1820
Formula for Cumulative Total Placements (144!/12!)/(4!)33
Cumulative Total Placements 3288577149009517628504635309346001109967738179693500072182535175905580380639255166212773768968230853960890313214534828900299747930204890570480387961372426696919207400000000000000000000000000000000

Groups Present 34
Unique Divisor 3
Formula for Cumulative Unique Divisor (36!/2!)
Cumulative Unique Divisor 185996663394950608733999724075417600000000
Formula for Unique Placements ((12!/8!)/(4!))/(3)
Unique Placement Possibilities 165
Formula For Cumulative Unique Placements ((144!/8!)/(4!)34)/(36!/2!)
Cumulative Unique Placements 8752015541822367857307331756170798386096154435809016956058384207071793879977213148527074938003002892875443283100206942904095602256739333269298076629638671875
Formula for Total Placements (12!/8!)/(4!)
Total Placement Possibilities 495
Formula for Cumulative Total Placements (144!/8!)/(4!)34
Cumulative Total Placements 1627845688759711266109794478126270549434040398948282535730354912073262288416431307275323015639274272710640705041194740345648375225451420832387792040879351214975007663000000000000000000000000000000000

Groups Present 35
Unique Divisor 2
Formula for Cumulative Unique Divisor (36!/1!)
Cumulative Unique Divisor 371993326789901217467999448150835200000000
Formula for Unique Placements ((8!/4!)/(4!))/(2)
Unique Placement Possibilities 35
Formula For Cumulative Unique Placements ((144!/4!)/(4!)35)/(36!/1!)
Cumulative Unique Placements 306320543963782875005756611465977943513365405253315593462043447247512785799202460198447622830105101250640514908507243001643346078985876664425432682037353515625
Formula for Total Placements (8!/4!)/(4!)
Total Placement Possibilities 70
Formula for Cumulative Total Placements (144!/4!)/(4!)35
Cumulative Total Placements 113949198213179785827685613468838938460382127926379777501124843845128360189150191509272611094749199689744849352883681821395386265781599458267145442861554585048250536410000000000000000000000000000000000

Groups Present 36
Unique Divisor 1
Formula for Cumulative Unique Divisor (36!/0!)
Cumulative Unique Divisor 371993326789901217467999448150835200000000
Formula for Unique Placements ((4!/0!)/(4!))/(1)
Unique Placement Possibilities 1
Formula For Cumulative Unique Placements ((144!/0!)/(4!)36)/(36!/0!)
Cumulative Unique Placements 306320543963782875005756611465977943513365405253315593462043447247512785799202460198447622830105101250640514908507243001643346078985876664425432682037353515625
Formula for Total Placements (4!/0!)/(4!)
Total Placement Possibilities 1
Formula for Cumulative Total Placements (144!/0!)/(4!)36
Cumulative Total Placements 113949198213179785827685613468838938460382127926379777501124843845128360189150191509272611094749199689744849352883681821395386265781599458267145442861554585048250536410000000000000000000000000000000000

=^.^=