Number of Possible Deals
In total there are 53,644,737,765,488,792,839,237,440,000 (5.36 x 1028) different deals possible, which is equal to . The immenseness of this number can be understood by answering the question "How large an area would you need to spread all possible bridge deals if each deal would occupy only one square millimeter?". The answer is: an area more than a hundred million times the total area of the earth.
Obviously, the deals that are identical except for swapping—say—the ♥2 and the ♥3 would be unlikely to give a different result. To make the irrelevance of small cards explicit (which is not always the case though), in bridge such small cards are generally denoted by an 'x'. Thus, the "number of possible deals" in this sense depends of how many non-honour cards (2, 3, .. 9) are considered 'indistinguishable'. For example, if 'x' notation is applied to all cards smaller than ten, then the suit distributions A987-K106-Q54-J32 and A432-K105-Q76-J98 would be considered identical.
The table below gives the number of deals when various numbers of small cards are considered indistinguishable.
| Suit composition | Number of deals |
|---|---|
| AKQJT9876543x | 53,644,737,765,488,792,839,237,440,000 |
| AKQJT987654xx | 7,811,544,503,918,790,990,995,915,520 |
| AKQJT98765xxx | 445,905,120,201,773,774,566,940,160 |
| AKQJT9876xxxx | 14,369,217,850,047,151,709,620,800 |
| AKQJT987xxxxx | 314,174,475,847,313,213,527,680 |
| AKQJT98xxxxxx | 5,197,480,921,767,366,548,160 |
| AKQJT9xxxxxxx | 69,848,690,581,204,198,656 |
| AKQJTxxxxxxxx | 800,827,437,699,287,808 |
| AKQJxxxxxxxxx | 8,110,864,720,503,360 |
| AKQxxxxxxxxxx | 74,424,657,938,928 |
| AKxxxxxxxxxxx | 630,343,600,320 |
| Axxxxxxxxxxxx | 4,997,094,488 |
| xxxxxxxxxxxxx | 37,478,624 |
Note that the last entry in the table (37,478,624) corresponds to the number of different distributions of the deck (the number of deals when cards are only distinguished by their suit).
Read more about this topic: Bridge Probabilities
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