With three players there are 807,723,671,040 different deals, and seven cards in hand have on average 35 points. After the deal at least one player holds a marriage in 64.8% of deals, and the king and queen of hearts are in a single hand in only 23% of deals. We calculated every number on this page exactly, by counting arrangements of the cards, and we give the method for each.
807,723,671,040 that many different deals with three players: 24! / (7! · 7! · 7! · 3!)
How many deals there are
We deal 24 cards: 7 to each of three players and 3 to the kitty (musik in Polish). The first player can get any 7 of the 24 cards, the second any 7 of the 17 that are left, the third any 7 of the remaining 10, and the last three go to the kitty. So the number of deals is C(24, 7) · C(17, 7) · C(10, 7) = 24! / (7! · 7! · 7! · 3!).
| What we count | How many |
|---|---|
| different hands of seven cards | 346,104 |
| different kitties of three cards | 2,024 |
| different deals with three players | 807,723,671,040 |
If you dealt one deal a minute, without sleep or breaks, going through all of them would take about 1.5 million years. And after a hundred thousand deals the chance that any of them has repeated is a mere 0.6% (the same reasoning as in the birthday problem).
How many points a hand has
The whole deck has 120 points in cards, so one card averages 5 points, and seven cards 35 points. The distribution is tightly packed around the mean: the standard deviation is 9.3 points, and the most frequent result is 33 points (4.8% of hands).
| A hand of seven cards has | Chance |
|---|---|
| from 30 to 40 points in cards | 44.2% |
| at most 20 points | 5.4% |
| at least 50 points | 6.5% |
| at least 60 points | 0.38% (1 in 265) |
| fewer than 10 points | 0.06% (1 in 1,633) |
The weakest possible hand is 6 points: four nines and three jacks. The strongest is 74: four aces and three tens. Remember that these are cards only, without marriages.
Nines and aces
There are four cards of each rank in the deck, so the chance that a seven-card hand contains exactly k of them (for example nines) is C(4, k) · C(20, 7 − k) / C(24, 7). The distribution is the same for nines, jacks, queens, kings, tens and aces.
| How many cards of a given rank in hand | Chance |
|---|---|
| 0 | 22.4% |
| 1 | 44.8% |
| 2 | 26.9% |
| 3 | 5.6% |
| 4 (four nines or four aces) | 0.33% (1 in 304) |
- Neither an ace nor a ten in the hand in 3.3% of deals (1 in 30). That is C(16, 7) / C(24, 7): seven cards chosen from the 16 that are neither aces nor tens.
- Four nines in anyone’s hand at the table: three hands at 0.33% each, and since two players cannot have them at once, 0.99% in all, that is once in 101 deals.
- In one game (we assume 25 deals, because that is how long a game lasts on average according to the simulation) someone at the table is dealt four nines with a chance of about 22%.
What is allowed then: four nines in the rules.
Marriages
How many marriages (king + queen pairs of one suit) does a seven-card hand have? We count by suit: for each suit we count in how many ways j of its 6 cards can be taken, with the king and queen together or apart, and we multiply the results for the four suits.
| Marriages in hand | Chance |
|---|---|
| 0 | 71.5% |
| 1 | 26.5% |
| 2 | 1.9% (1 in 52) |
| 3 | 0.021% (1 in 4,807) |
So at least one marriage is 28.5%, and this agrees with the figure on the marriages page, which was taken from the game engine.
The whole deal at once
With three players the cards go to four “groups”: three hands of 7 cards and the kitty of 3.
- The king and queen of hearts in the same place. Two specific cards land in the same group with chance (C(7, 2) + C(7, 2) + C(7, 2) + C(3, 2)) / C(24, 2) = 66/276. Of this, 63/276, that is 22.8%, is one player’s hand, and 3/276, that is 1.1%, is the kitty. 76.1% of deals split this pair.
- At least one marriage in any hand (not in the kitty): 64.8%. In 35.2% of deals nobody at the table holds any marriage after the deal. We count this by checking all 65,536 ways of placing the eight kings and queens in four groups and, for each, counting the ways to add the remaining cards.
Since pairs are so often split, it matters that the kitty can complete the missing half: bidding and the kitty.
The kitty
The kitty is three cards out of 24, so on average 15 points in cards (3 × 5).
| In the kitty | Chance |
|---|---|
| there is at least one ace | 43.7% |
| there is an ace or a ten | 72.3% |
| there are at least 20 points | 24.9% |
| there are at most 5 points | 8.1% |
| there are three nines | 0.20% (1 in 506) |
The most that can lie in the kitty is 33 points (three aces). The chance of at least one ace is 1 − C(20, 3) / C(24, 3): from one we subtract the chance that all three cards were chosen from the 20 cards that are not aces.
How we counted
- Exactly, not from a sample. None of the numbers here comes from random draws. It is all combinatorics on whole numbers (BigInt), divided only at the end.
- Three checks. The results agree (to five decimal places) with the data that the game engine calculated for the page Thousand statistics, with going through all 346,104 hands one by one, and with a second, independent method: 200,000 shuffled deals.
- The script. The code is the file
liczby.mjsin the repository of the site (npm run liczby); it recalculates the data of this page, and a test makes sure that the numbers on the page always agree with the script.
How to cite
Thousand (1000), “The game in numbers”, as of 17 October 2026 — with the address of this page.