Counting Bridge Deals
by Jeroen Warmerdam
10 March 2003
The number of bridge deals
can be calculated with a simple formula: 52! / (13!)^{4}.
On the newsgroup rec.games.bridge someone asked to count the
number of deals if we consider the rank of small cards irrelevant. In other
words, we deal a new deck consisting of the four normal suits and each suit has
AKQJT and eight indistinguishable cards called x. How many deals are there now?
It seems there is no simple formula for this number of
deals. I solved it with the assistance of a computer program. The answer to the
question above is 800.827.437.699.287.808. This table also gives the number of
deals when we consider less or more cards indistinguishable.
Highest small card |
Number of deals |
2 |
53.644.737.765.488.792.839.237.440.000 |
3 |
7.811.544.503.918.790.990.995.915.520 |
4 |
445.905.120.201.773.774.566.940.160 |
5 |
14.369.217.850.047.151.709.620.800 |
6 |
314.174.475.847.313.213.527.680 |
7 |
5.197.480.921.767.366.548.160 |
8 |
69.848.690.581.204.198.656 |
9 |
800.827.437.699.287.808 |
10 |
8.110.864.720.503.360 |
J |
74.424.657.938.928 |
Q |
630.343.600.320 |
K |
4.997.094.488 |
A |
37.478.624 |
If you are interested in the method I used, you can
download the Source
code. The table was generated within one minute.