## Poll

3 votes (60%) | |||

1 vote (20%) | |||

No votes (0%) | |||

No votes (0%) | |||

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1 vote (20%) |

**5 members have voted**

October 5th, 2019 at 8:01:11 PM
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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?

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.

October 5th, 2019 at 8:23:07 PM
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7 x 12 = 84.

October 5th, 2019 at 9:43:14 PM
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the concept is called 'arbitrary precision' and free stuff I use is Pari/Gp and (for windows) Precise CalculatorQuote:WizardHowever, I think others do have access to software I don't that are not so limited.

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)

October 5th, 2019 at 9:52:42 PM
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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.

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.

October 6th, 2019 at 12:34:07 PM
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A standard 10 inch slide rule will give you the answer to 3 digits. Make an 80 inch one and you’ll get 24 digits.

It’s all about making that GTA

October 6th, 2019 at 3:26:17 PM
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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).

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).

October 7th, 2019 at 5:57:29 AM
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Thanks all for your suggestions.

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?

Quote:ThatDonGuyMicrosoft .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.

October 7th, 2019 at 6:31:49 PM
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yes.Quote:WizardCan anyone provide the EXACT number of combinations for any total. If so, how did you do it?

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)

October 7th, 2019 at 7:16:22 PM
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Thanks for the offer, but I really want to do this myself.

Do any forum members have Mathematica?

Do any forum members have Mathematica?

It's not whether you win or lose; it's whether or not you had a good bet.

October 7th, 2019 at 7:33:13 PM
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cool, at least you can compare your results to mine (the posted one) IF you want, once you complete them.Quote:WizardThanks for the offer, but I really want to do this myself.

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)