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.