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5 members have voted

Wizard
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Wizard
Joined: Oct 14, 2009
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October 5th, 2019 at 8:01:11 PM permalink
If there is a heaven, in the unlikely event I'm sent there, I hope there are compilers and spreadsheets that can handle an unlimited number of digits. However, for now, I'm limited to integers as large as 2^64. However, I think others do have access to software I don't that are not so limited.

That said, here is the question at hand. Take an 8-deck shoe. Cards are scored as follows:

A = 1
2-10 = pip value
J = 11
Q = 12
K = 13

What is the number of combination for any given total of 12 random cards, chosen without replacement?

Unfortunately, the combinations ran a little over 2^64, so I had to revert to a decimal.

Can anyone provide the EXACT number of combinations for any total. If so, how did you do it?

Total Combinations
156 225,792,840
155 4,128,783,360
154 36,126,854,400
153 209,311,365,120
152 925,616,009,760
151 3,373,898,151,936
150 10,623,140,027,904
149 29,809,235,896,320
148 76,197,335,731,008
147 180,296,163,767,296
146 399,724,519,660,032
145 838,212,685,748,224
144 1,674,991,374,949,480
143 3,208,902,130,674,680
142 5,922,867,556,960,760
141 10,575,845,958,791,100
140 18,331,035,981,153,100
139 30,930,773,414,340,600
138 50,930,223,409,649,600
137 82,003,429,211,543,500
136 129,336,417,589,173,000
135 200,122,672,056,249,000
134 304,176,175,625,480,000
133 454,676,126,820,014,000
132 669,055,131,415,173,000
131 970,038,813,588,301,000
130 1,386,839,163,709,630,000
129 1,956,496,319,319,710,000
128 2,725,353,970,931,820,000
127 3,750,642,161,501,160,000
126 5,102,128,424,591,150,000
125 6,863,784,020,935,050,000
124 9,135,397,420,885,390,000
123 12,034,052,413,826,400,000
122 15,695,374,428,937,700,000
121 20,274,436,431,451,800,000
120 25,946,206,540,362,800,000
119 32,905,414,042,970,200,000
118 41,365,710,559,273,000,000
117 51,558,009,555,994,300,000
116 63,727,901,432,043,100,000
115 78,132,062,984,093,600,000
114 95,033,609,735,335,800,000
113 114,696,376,221,781,000,000
112 137,378,153,142,167,000,000
111 163,322,958,869,565,000,000
110 192,752,475,839,657,000,000
109 225,856,836,660,953,000,000
108 262,784,999,186,317,000,000
107 303,635,000,181,527,000,000
106 348,444,421,949,181,000,000
105 397,181,440,164,700,000,000
104 449,736,842,333,641,000,000
103 505,917,411,150,574,000,000
102 565,441,055,054,571,000,000
101 627,934,037,529,866,000,000
100 692,930,608,646,902,000,000
99 759,875,275,848,309,000,000
98 828,127,869,827,542,000,000
97 896,971,465,737,049,000,000
96 965,623,115,097,711,000,000
95 1,033,247,231,547,180,000,000
94 1,098,971,361,130,320,000,000
93 1,161,903,957,936,980,000,000
92 1,221,153,687,878,860,000,000
91 1,275,849,700,278,110,000,000
90 1,325,162,245,345,500,000,000
89 1,368,322,977,053,450,000,000
88 1,404,644,269,407,700,000,000
87 1,433,536,888,657,070,000,000
86 1,454,525,407,107,490,000,000
85 1,467,260,812,125,590,000,000
84 1,471,529,856,918,220,000,000
83 1,467,260,812,125,600,000,000
82 1,454,525,407,107,490,000,000
81 1,433,536,888,657,070,000,000
80 1,404,644,269,407,700,000,000
79 1,368,322,977,053,450,000,000
78 1,325,162,245,345,500,000,000
77 1,275,849,700,278,110,000,000
76 1,221,153,687,878,860,000,000
75 1,161,903,957,936,980,000,000
74 1,098,971,361,130,320,000,000
73 1,033,247,231,547,180,000,000
72 965,623,115,097,710,000,000
71 896,971,465,737,047,000,000
70 828,127,869,827,541,000,000
69 759,875,275,848,307,000,000
68 692,930,608,646,901,000,000
67 627,934,037,529,864,000,000
66 565,441,055,054,572,000,000
65 505,917,411,150,574,000,000
64 449,736,842,333,641,000,000
63 397,181,440,164,700,000,000
62 348,444,421,949,181,000,000
61 303,635,000,181,527,000,000
60 262,784,999,186,317,000,000
59 225,856,836,660,953,000,000
58 192,752,475,839,657,000,000
57 163,322,958,869,565,000,000
56 137,378,153,142,167,000,000
55 114,696,376,221,781,000,000
54 95,033,609,735,335,800,000
53 78,132,062,984,093,600,000
52 63,727,901,432,043,100,000
51 51,558,009,555,994,300,000
50 41,365,710,559,273,100,000
49 32,905,414,042,970,200,000
48 25,946,206,540,362,800,000
47 20,274,436,431,451,800,000
46 15,695,374,428,937,700,000
45 12,034,052,413,826,400,000
44 9,135,397,420,885,390,000
43 6,863,784,020,935,050,000
42 5,102,128,424,591,150,000
41 3,750,642,161,501,160,000
40 2,725,353,970,931,820,000
39 1,956,496,319,319,710,000
38 1,386,839,163,709,630,000
37 970,038,813,588,301,000
36 669,055,131,415,173,000
35 454,676,126,820,014,000
34 304,176,175,625,480,000
33 200,122,672,056,249,000
32 129,336,417,589,173,000
31 82,003,429,211,543,500
30 50,930,223,409,649,600
29 30,930,773,414,340,600
28 18,331,035,981,153,100
27 10,575,845,958,791,100
26 5,922,867,556,960,760
25 3,208,902,130,674,680
24 1,674,991,374,949,480
23 838,212,685,748,224
22 399,724,519,660,032
21 180,296,163,767,296
20 76,197,335,731,008
19 29,809,235,896,320
18 10,623,140,027,904
17 3,373,898,151,936
16 925,616,009,760
15 209,311,365,120
14 36,126,854,400
13 4,128,783,360
12 225,792,840
Total 47,778,959,505,588,100,000,000
It's not whether you win or lose; it's whether or not you had a good bet.
ChumpChange
ChumpChange
Joined: Jun 15, 2018
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October 5th, 2019 at 8:23:07 PM permalink
7 x 12 = 84.
7craps
7craps
Joined: Jan 23, 2010
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October 5th, 2019 at 9:43:14 PM permalink
Quote: Wizard

However, I think others do have access to software I don't that are not so limited.

the concept is called 'arbitrary precision' and free stuff I use is Pari/Gp and (for windows) Precise Calculator
for example:
my Excel's combin(416,12) = 47,778,959,505,588,400,000,000

pari/gp: gp > binomial(416,12)
%1 = 47778959505588340858920
pcalc: 416nCr12
47 778 959 505 588 340 858 920
no problem there

C++ has
C++ BigInt Class (never used it, but know some that have and they never complained about using it)
https://en.wikipedia.org/wiki/List_of_arbitrary-precision_arithmetic_software

try starting there and I am certain one will come by and show their results. I just do not have time to look at it
winsome johnny (not Win some johnny)
gordonm888
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gordonm888 
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October 5th, 2019 at 9:52:42 PM permalink
Well, you have used combination math to calculate each of those numbers - call them x(n) -in the table. Go back to the formulas that you used, and express each and any of the numbers x(n) as a product of two large numbers, i.e.,

x(n) = A * B where A and B are both large numbers for which all the digits can be calculated on your spreadsheet.

Then, for any given A and B, simply multiply the two numbers by hand, this will give you all the digits.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
Ace2
Ace2
Joined: Oct 2, 2017
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October 6th, 2019 at 12:34:07 PM permalink
A standard 10 inch slide rule will give you the answer to 3 digits. Make an 80 inch one and youíll get 24 digits.
Itís all about making that GTA
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
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October 6th, 2019 at 3:26:17 PM permalink
Microsoft .NET has a BigInteger class that allows for unlimited precision integers.

It is not loaded by default; in Visual Studio, you have to add the System.Numerics namespace manually (through the "Add Reference" option in the menu).
Wizard
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Wizard
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October 7th, 2019 at 5:57:29 AM permalink
Thanks all for your suggestions.

Quote: ThatDonGuy

Microsoft .NET has a BigInteger class that allows for unlimited precision integers.



I had a look at this, but don't trust myself to implement it properly.

I think I'll have to just get past this for now and continue living in small-number purgatory.

I'm still using Visual Studio 2005. I am long overdue to get a new computer. Any suggestions on a more modern C++ compiler that can handle integers of unlimited size?
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
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October 7th, 2019 at 6:31:49 PM permalink
Quote: Wizard

Can anyone provide the EXACT number of combinations for any total. If so, how did you do it?

yes.
I use a generating function in pari/gp
b=(prod(i=1, 13, 1 + x*y^i))^32;
print(b)

took a few seconds to complete (i have not learned how to just get the coefficient of x^12
it returns x^1 to x^416)
to write to a text file
so a text file works for me
results
y^156 is the sum of 156
+ 225792840*y^156 
+ 4128783360*y^155
+ 36126854400*y^154
+ 209311365120*y^153
+ 925616009760*y^152
+ 3373898151936*y^151
+ 10623140027904*y^150
+ 29809235896320*y^149
+ 76197335731008*y^148
+ 180296163767296*y^147
+ 399724519660032*y^146
+ 838212685748224*y^145
+ 1674991374949480*y^144
+ 3208902130674688*y^143
+ 5922867556960768*y^142
+ 10575845958791168*y^141
+ 18331035981153120*y^140
+ 30930773414340608*y^139
+ 50930223409649664*y^138
+ 82003429211543552*y^137
+ 129336417589173856*y^136
+ 200122672056249344*y^135
+ 304176175625480960*y^134
+ 454676126820014080*y^133
+ 669055131415173288*y^132
+ 970038813588301824*y^131
+ 1386839163709634048*y^130
+ 1956496319319717888*y^129
+ 2725353970931822208*y^128
+ 3750642161501161472*y^127
+ 5102128424591154688*y^126
+ 6863784020935055360*y^125
+ 9135397420885393056*y^124
+ 12034052413826405376*y^123
+ 15695374428937774848*y^122
+ 20274436431451837440*y^121
+ 25946206540362835400*y^120
+ 32905414042970269696*y^119
+ 41365710559273110016*y^118
+ 51558009555994372096*y^117
+ 63727901432043160000*y^116
+ 78132062984093763584*y^115
+ 95033609735335922432*y^114
+ 114696376221781665792*y^113
+ 137378153142167576000*y^112
+ 163322958869565643776*y^111
+ 192752475839657541120*y^110
+ 225856836660953043968*y^109
+ 262784999186317858568*y^108
+ 303635000181527090176*y^107
+ 348444421949183178496*y^106
+ 397181440164700567552*y^105
+ 449736842333643290048*y^104
+ 505917411150573856768*y^103
+ 565441055054573899776*y^102
+ 627934037529867206656*y^101
+ 692930608646904309728*y^100
+ 759875275848307115008*y^99
+ 828127869827542576640*y^98
+ 896971465737048110080*y^97
+ 965623115097710930408*y^96
+ 1033247231547180078080*y^95
+ 1098971361130330388224*y^94
+ 1161903957936987926528*y^93
+ 1221153687878866101472*y^92
+ 1275849700278120556544*y^91
+ 1325162245345502497792*y^90
+ 1368322977053461135360*y^89
+ 1404644269407710191808*y^88
+ 1433536888657081828352*y^87
+ 1454525407107492726528*y^86
+ 1467260812125610382336*y^85
+ 1471529856918235043336*y^84
+ 1467260812125610382336*y^83
+ 1454525407107492726528*y^82
+ 1433536888657081828352*y^81
+ 1404644269407710191808*y^80
+ 1368322977053461135360*y^79
+ 1325162245345502497792*y^78
+ 1275849700278120556544*y^77
+ 1221153687878866101472*y^76
+ 1161903957936987926528*y^75
+ 1098971361130330388224*y^74
+ 1033247231547180078080*y^73
+ 965623115097710930408*y^72
+ 896971465737048110080*y^71
+ 828127869827542576640*y^70
+ 759875275848307115008*y^69
+ 692930608646904309728*y^68
+ 627934037529867206656*y^67
+ 565441055054573899776*y^66
+ 505917411150573856768*y^65
+ 449736842333643290048*y^64
+ 397181440164700567552*y^63
+ 348444421949183178496*y^62
+ 303635000181527090176*y^61
+ 262784999186317858568*y^60
+ 225856836660953043968*y^59
+ 192752475839657541120*y^58
+ 163322958869565643776*y^57
+ 137378153142167576000*y^56
+ 114696376221781665792*y^55
+ 95033609735335922432*y^54
+ 78132062984093763584*y^53
+ 63727901432043160000*y^52
+ 51558009555994372096*y^51
+ 41365710559273110016*y^50
+ 32905414042970269696*y^49
+ 25946206540362835400*y^48
+ 20274436431451837440*y^47
+ 15695374428937774848*y^46
+ 12034052413826405376*y^45
+ 9135397420885393056*y^44
+ 6863784020935055360*y^43
+ 5102128424591154688*y^42
+ 3750642161501161472*y^41
+ 2725353970931822208*y^40
+ 1956496319319717888*y^39
+ 1386839163709634048*y^38
+ 970038813588301824*y^37
+ 669055131415173288*y^36
+ 454676126820014080*y^35
+ 304176175625480960*y^34
+ 200122672056249344*y^33
+ 129336417589173856*y^32
+ 82003429211543552*y^31
+ 50930223409649664*y^30
+ 30930773414340608*y^29
+ 18331035981153120*y^28
+ 10575845958791168*y^27
+ 5922867556960768*y^26
+ 3208902130674688*y^25
+ 1674991374949480*y^24
+ 838212685748224*y^23
+ 399724519660032*y^22
+ 180296163767296*y^21
+ 76197335731008*y^20
+ 29809235896320*y^19
+ 10623140027904*y^18
+ 3373898151936*y^17
+ 925616009760*y^16
+ 209311365120*y^15
+ 36126854400*y^14
+ 4128783360*y^13
+ 225792840*y^12


I know this does NOT help you out with the process
I am a terrible teacher, I just know why it works and how to do it in pari/gp

good luck
back to wsop.com

if you like
I could save the complete text file (I deleted it already) that shows all the ways for drawing 1 to 416 inclusive from 8 decks
Last edited by: 7craps on Oct 7, 2019
winsome johnny (not Win some johnny)
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
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October 7th, 2019 at 7:16:22 PM permalink
Thanks for the offer, but I really want to do this myself.

Do any forum members have Mathematica?
It's not whether you win or lose; it's whether or not you had a good bet.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
October 7th, 2019 at 7:33:13 PM permalink
Quote: Wizard

Thanks for the offer, but I really want to do this myself.

cool, at least you can compare your results to mine (the posted one) IF you want, once you complete them.

I think I did this correctly...
generating function is a simple formula
(complete text file took 'last result computed in 19,875 ms.' )

good luck

added: for the sum=88
Pari :1,404,644,269,407,710,191,808
Wiz: 1,404,644,269,407,700,000,000

also: the code could just count the NUMBER of 'unique combinations'
there are only 2,704,156 of those. from there, the total number of combinations is simple math

... 9657700*x^14 + 5200300*x^13 + 2704156*x^12 + 1352078*x^11 + 646646*x^10 + 293930*x^9 + 125970*x^8 + 50388*x^7 + 18564*x^6 + 6188*x^5 + 1820*x^4 + 455*x^3 + 91*x^2 + 13*x + 1
this part result is from the expansion of this: print((1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24+x^25+x^26+x^27+x^28+x^29+x^30+x^31+x^32)^13)

also added: text file for drawing 0 to 20 cards without replacement. (72 KB in size)
combinations for all sums (y coefficient is the sum and the x coefficient is the number of cards drawn)
they have their own line in notepad++ (line #9 is for draw 12 cards)

https://drive.google.com/open?id=1zEi298vpVtQw-yGA-7OKl-fqYWvJW0Qh

enjoy generating functions
Last edited by: 7craps on Oct 8, 2019
winsome johnny (not Win some johnny)

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