Start with the assumptions
This model deals three distinct cards uniformly from a standard 52-card deck. It uses the six ordinary high-hand categories, accepts A-2-3 as a run and includes twelve distinct three-rank runs. Wild cards and special deck-selection rules belong to their own models.
The order assigned to A-2-3 changes comparisons, while its presence as an accepted run leaves these category totals the same.
The complete distribution
| Category | Distinct hands | Share of the model |
|---|---|---|
| Trail | 52 | 0.2353% |
| Pure sequence | 48 | 0.2172% |
| Sequence | 720 | 3.2579% |
| Color | 1,096 | 4.9593% |
| Pair | 3,744 | 16.9412% |
| High card | 16,440 | 74.3891% |
| Complete model | 22,100 | 100% |
Percentages are rounded for display. The category counts sum exactly to 22,100, so each possible hand belongs to one row.
Build the counts step by step
- All hands: choose any three of fifty-two distinct cards: C(52,3) = 22,100.
- Trail: choose one of thirteen ranks and three of its four suits: 13 × C(4,3) = 52.
- Pure sequence: twelve rank runs in each of four suits: 12 × 4 = 48.
- Sequence: for each run, assign suits in 4³ ways and separate the four same-suit choices: 12 × (64 − 4) = 720.
- Color: choose three ranks in one suit and separate its twelve runs: 4 × (C(13,3) − 12) = 1,096.
- Pair: choose the paired rank and its two suits, then a different kicker rank and suit: 13 × 6 × 12 × 4 = 3,744.
- High card: the remaining hands total 16,440.
Frequency and ranking are different ideas
Pure sequences number 48 in this model; trails number 52. Yet the chosen category ladder places trail above pure sequence. The ladder is a game rule, while the count comes from the deck model.
This distinction also helps with a single example. A frequency describes a category across many possible deals. The hand comparison tool evaluates the category and ranks of the cards entered.
Use the model carefully
A known card, a restricted deck or a wild-card procedure changes the question being counted. State those conditions before carrying a percentage into another format.
For a useful learning exercise, name the category of ten sample hands, then compare your classifications with the chart. The practice tool focuses on recognition; this table explains the complete distribution behind the defined model.