In a uniformly dealt, standard 52-card deck without jokers, there are 22,100 different three-card hands when order does not matter. Counting their categories describes how often a hand can occur in that model. It does not give your chance of winning a table, prove an app is fair or suggest a profitable betting method.
Define the rules before calculating
The classification here follows the traditional categories described by John McLeod's Teen Patti rules: trail, same-suit sequence, other sequence, colour, pair and high card. Both ace-two-three and queen-king-ace count as runs; king-ace-two does not. Jokers, extra cards and changed classifications require a different calculation. This is not an audit of an online operator.
Cards have both a rank and a suit. Three aces of different suits are different physical cards. Conversely, dealing the same three cards in another order does not create another hand for this count. That is why 52 times 51 times 50 must be divided by the six possible orders of each set: the result is 22,100, not 132,600.
Count each hand in exactly one category
The following figures are our enumeration of that defined model, not measured app results. Percentages are rounded to three decimal places. Categories are mutually exclusive: a same-suit run belongs to pure sequence, not also to colour and ordinary sequence.
| Category | Distinct hands | Model probability |
|---|---|---|
| Trail | 52 | 0.235% |
| Pure sequence | 48 | 0.217% |
| Other sequence | 720 | 3.258% |
| Colour, excluding pure sequences | 1,096 | 4.959% |
| Pair | 3,744 | 16.941% |
| High card | 16,440 | 74.389% |
For a trail, choose one of thirteen ranks and three of its four suits: thirteen times four gives 52. For a pair, choose its rank, two of its four suits, a different rank and one suit for the odd card. That gives thirteen times six times twelve times four, or 3,744.
There are twelve accepted three-rank runs in this model. Each has four same-suit arrangements, giving 48 pure sequences. Each run has sixty-four suit arrangements in total; removing the four same-suit arrangements leaves sixty. Twelve times sixty gives 720 ordinary sequences.
For colour, choose three different ranks from thirteen and one shared suit. There are 286 rank combinations, or 1,144 same-suit hands across four suits. Subtract the 48 pure sequences to leave 1,096. Everything outside the first five categories belongs to high card: subtracting their combined 5,660 hands from 22,100 leaves 16,440.
Check the arithmetic independently
A reproducible check is to label the deck's cards from zero through fifty-one, then visit every triple of distinct labels with the first smaller than the second and the second smaller than the third. Assign each triple to its first matching category in the table's order. The resulting six counts must add to 22,100. Our exhaustive check produces the same totals as the formulas above.
If your count is six times too large, you probably counted different dealing orders separately. If the total exceeds 22,100 by a smaller amount, check whether pure sequences were also counted as colours or ordinary runs. If it is smaller, inspect how ace-low runs and high-card leftovers were classified.
Rarity is not ranking, and a hand is not a win
A pure sequence has fewer combinations than a trail in this model, yet the stated rules rank trail higher. Hand order is a rule, not something that can always be reconstructed by sorting category frequencies.
Likewise, the pair percentage is not a pair's winning percentage. An outcome comparison would need the other hands and the applicable rules; the other hands also come from the remaining deck rather than independent fresh decks. These calculations do not include decisions, charges or settlement. Use them to check a probability explanation, not to predict the next deal or decide how much to stake.