Contract Bridge

Or Hearts, or Spades or any other game where a deck of 52 cards consisting of 13 cards in 4 suits is fully dealt to 4 players.

09 Apr 07

Permutations, Formulas, Computations, Odds

Note - This is what got me started on this project in the first place. I used to play a lot of duplicate contract bridge and at one time became a member of the American Contract Bridge League (ACBL). I remember reading where they stated that there was the possibility of 158,753,389,900 different hands. For some reason, this was an understatement by a factor of four. See below.

Formula for total number of combinations: (52!/39!)/(13!) = 635,013,559,600

Number of possible combinations of cards in various sized suits:

13 card suit - (13!/0!)/(13!) = 1
12 card suit - (13!/1!)/(12!) = 13
11 card suit - (13!/2!)/(11!) = 78
10 card suit - (13!/3!)/(10!) = 286
9 card suit - (13!/4!)/(9!) = 715
8 card suit - (13!/5!)/(8!) = 1287
7 card suit - (13!/6!)/(7!) = 1716
6 card suit - (13!/7!)/(6!) = 1716
5 card suit - (13!/8!)/(5!) = 1287
4 card suit - (13!/9!)/(4!) = 715
3 card suit - (13!/10!)/(3!) = 286
2 card suit - (13!/11!)/(2!) = 78
1 card suit - (13!/12!)/(1!) = 13
Void - (13!/13!!)/(0!) = 1

Proof of this is offered in the Suit Content By Distribution Chart on this site - link below.

Note - These numbers are consistent with the numbers contained in Paschal's Triangle at this level.

Obviously, the number of cards in each hand will total 13 in all instances. In the examples below, for instance, a 13-0-0-0 distribution had 4 permutations - either 13 spades, 13 hearts, 13 diamonds or 13 clubs, and each of these combinations, whether 13 or 0, has only 1 possibility. Therefore, there are only 4 possible hands with that distribution. Also, with the same number of cards in 3 suits, regardless of the number of cards in each suit, there are only 4 distributional permutations. A hand with the same number of cards in two suits has 12 permutations. A hand with a different number of cards in each suit, such as a 1-3-4-5 distribution, has 24 permutations.

Given the above factors, the number of possible hands for each distribution is:

Distribution	Permutations	Formula	Possibilities
0- 0- 0-13	4	1 x 1 x 1 x 1	4
0- 0- 1-12	12	1 x 1 x 13 x 13	2,028
0- 0- 2-11	12	1 x 1 x 78 x 78	73,008
0- 0- 3-10	12	1 x 1 x 286 x 286	981,552
0- 0- 4- 9	12	1 x 1 x 715 x 715	6,134,700
0- 0- 5- 8	12	1 x 1 x 1,287 x 1,287	19,876,428
0- 0- 6- 7	12	1 x 1 x 1,716 x 1,716	35,335,872
0- 1- 1-11	12	1 x 13 x 13 x 78	158,184
0- 1- 2-10	24	1 x 13 x 78 x 286	6,960,096
0- 1- 3- 9	24	1 x 13 x 286 x 715	63,800,880
0- 1- 4- 8	24	1 x 13 x 715 x 1,287	287,103,960
0- 1- 5- 7	24	1 x 13 x 1,287 x 1,716	689,049,504
0- 1- 6- 6	12	1 x 13 x 1,716 x 1,716	459,336,336
0- 2- 2- 9	12	1 x 78 x 78 x 715	52,200,720
0- 2- 3- 8	24	1 x 78 x 286 x 1,287	689,049,504
0- 2- 4- 7	24	1 x 78 x 715 x 1,716	2,296,831,680
0- 2- 5- 6	24	1 x 78 x 1,287 x 1,716	4,134,297,024
0- 3- 3- 7	12	1 x 286 x 286 x 1,716	1,684,343,232
0- 3- 4- 6	24	1 x 286 x 715 x 1,716	8,421,716,160
0- 3- 5- 5	12	1 x 286 x 1,287 x 1,287	5,684,658,408
0- 4- 4- 5	12	1 x 715 x 715 x 1,287	7,895,358,900
1- 1- 1-10	4	13 x 13 x 13 x 286	2,513,368
1- 1- 2- 9	12	13 x 13 x 78 x 715	113,101,560
1- 1- 3- 8	12	13 x 13 x 286 x 1,287	746,470,296
1- 1- 4- 7	12	13 x 13 x 715 x 1,716	2,488,234,320
1- 1- 5- 6	12	13 x 13 x 1,287 x 1,716	4,478,821,776
1- 2- 2- 8	12	13 x 78 x 78 x 1,287	1,221,496,848
1- 2- 3- 7	24	13 x 78 x 286 x 1,716	11,943,524,736
1- 2- 4- 6	24	13 x 78 x 715 x 1,716	29,858,811,840
1- 2- 5- 5	12	13 x 78 x 1,287 x 1,287	20,154,697,992
1- 3- 3- 6	12	13 x 286 x 286 x 1,716	21,896,462,016
1- 3- 4- 5	24	13 x 286 x 715 x 1,287	82,111,732,560
1- 4- 4- 4	4	13 x 715 x 715 x 715	19,007,345,500
2- 2- 2- 7	4	78 x 78 x 78 x 1,716	3,257,324,928
2- 2- 3- 6	12	78 x 78 x 286 x 1,716	35,830,574,208
2- 2- 4- 5	12	78 x 78 x 715 x 1,287	67,182,326,640
2- 3- 3- 5	12	78 x 286 x 286 x 1,287	98,534,079,072
2- 3- 4- 4	12	78 x 286 x 715 x 715	136,852,887,600
3- 3- 3- 4	4	286 x 286 x 286 x 715	66,905,856,160

Total = 635,013,559,600

Note that the most popular non-suit-specific hand distribution is the 4-4-3-2 distribution. The second most popular distribution is the 5-3-3-2, distribution, followed by the 5-4-3-1 distribution, then the 5-4-2-2 distribution. The 4-3-3-3 distribution, seemingly the most popular, is only 5th on the list. Odds of getting this distribution in any one hand are approximately 1::9.49.

When all this is said and done, there remain 39 cards to be dealt to the other three hands.

Formula for the number of possible combinations in the second hand: (39!/26!)/(13!) = 8,122,425,444
Formula for the number of possible combinations in the third hand: (26!/13!)/(13!) = 10,400,600

The fourth hand is dealt the remaining 13 cards and therefore has only one possibility.

Given this, the number of possibilities at the table for each and every deal can be expressed as follows:

(52!)/((13!)⁴) = 53,644,737,765,488,792,839,237,440,000

Given this, if one hand is dealt 13 specific cards, the total number of possibilities for the other three hands can be expressed as follows:

(39!)/((13!)³) = 84,478,098,072,866,400

Distributional Possibilities

(Author's Note: Don't know just how far this section is going to go.....)

Given the above information it can be calculated that there are 560 suit-specific distributional possibilities in each hand. By calculation, there are 37,478,624 distributional possibilities for the four hands at each deal, starting from the 24 possibilities of having 13 of one suit in all four hands and ranging all the way to the 96,537 possibilities when one hand has 4-3-3-3 distribution given that the 4-card suit is a specific suit. Even with 13 cards of a specific suit in one hand there are 5,565 total distributional possibilities in the other three hands.

I believe I have a program capable of generating a database and feeding all the possibilities into it from which a complete chart could be constructed, but it would be huge. I may add to this one from time to time but I doubt that this will ever be complete. The files would just be too large.

Links

Distribution Chart - 00000001 - 00010000.
Distribution Chart - 20000001 - 20010000.
Distribution Chart - 20010001 - 20020000.
Distribution Chart - 20020001 - 20030000.
Distribution Chart - 20030001 - 20040000.
Distribution Chart - 20040001 - 20050000.
Distribution Chart - 20050001 - 20060000.
Distribution Chart - 20060001 - 20070000.
Distribution Chart - 20070001 - 20080000.
Distribution Chart - 20080001 - 20090000.
Distribution Chart - 20090001 - 20100000.
Distribution Chart - 20100001 - 20110000.
Distribution Chart - 20110001 - 20120000.
Distribution Chart - 20120001 - 20130000.
Distribution Chart - 20130001 - 20140000.
Distribution Chart - 20140001 - 20150000.
Distribution Chart - 20150001 - 20160000.
Distribution Chart - 20160001 - 20170000.
Distribution Chart - 20170001 - 20180000.
Distribution Chart - 20180001 - 20190000.
Distribution Chart - 20190001 - 20200000.
Distribution Chart - 20200001 - 20210000.
Distribution Chart - 20210001 - 20220000.
Distribution Chart - 20220001 - 20230000.
Distribution Chart - 20230001 - 20240000.
Distribution Chart - 20240001 - 20250000.
Distribution Chart - 20250001 - 20260000.
Distribution Chart - 20260001 - 20270000.
Distribution Chart - 20270001 - 20280000.
Distribution Chart - 20280001 - 20290000.
Distribution Chart - 20290001 - 20300000.
Distribution Chart - 20300001 - 20310000.
Distribution Chart - 20310001 - 20320000.
Distribution Chart - 20320001 - 20330000.
Distribution Chart - 20330001 - 20340000.
Distribution Chart - 20340001 - 20350000.
Distribution Chart - 20350001 - 20360000.
Distribution Chart - 20360001 - 20370000.
Distribution Chart - 20370001 - 20380000.
Distribution Chart - 20380001 - 20390000.
Distribution Chart - 20390001 - 20400000.
Distribution Chart - 20400001 - 20410000.
Distribution Chart - 20410001 - 20420000.
Distribution Chart - 20420001 - 20430000.
Distribution Chart - 20430001 - 20440000.
Distribution Chart - 20440001 - 20450000.
Distribution Chart - 20450001 - 20460000.
Distribution Chart - 20460001 - 20470000.
Distribution Chart - 20470001 - 20480000.
Distribution Chart - 20480001 - 20490000.
Distribution Chart - 20490001 - 20500000.

Hand Chart - 000000000001 - 000000100000.
Hand Chart - 000001900001 - 000002000000.

Note: Distribution and hand charts between the lowest and highest numbers listed above have been previously posted but have been removed from the site due to site size limitations. They are available and will be reposted upon request.

Note: A new, more efficient method of distribution data generation and processing has been initiated. All distribution charts are reproduced with slightly different formatting and will remain posted for about two weeks. Additional hand charts have been generated but await processing.

Suit Content By Distribution.

Ultimate Bridge Chart Demo.

For approaches from a different angle, particularly dealing with distributional percentages for one and two hands, look here.

Progress

Description	Total	Percentage
Possible Distributions	37478624	100.000000
Distributions Generated	22048462	58.829433
Distributions Processed	21000000	56.031940
Distributions Posted	20500000	54.697846
Possible Hands	635013559600	100.000000
Hands Generated	53524680	0.008429
Hands Posted	2000000	0.000315

In Closing

The ultimate bridge chart (or table), of course, would display all combinations of cards at all positions, along with the hand number for each hand at each position. I feel safe in saying no sane person would attempt this, given the numbers shown above. Even if one were to try, the storage requirements, even for the chart, would be staggering; something on the order of 5 followed by 30 zeros (5X10³⁰) bytes. However, if one feels like proving me wrong, the start of the chart and certain milestones would look something like you can see in the Demo link above.

=^.^=