54s: the 4 combos
For 54s, the five and the four must share a suit. With 4 suits and one way to pair them in each, the count is 4.
Every 54s you can be dealt is one of these 4 holdings. They are listed once each, higher card first.
How cards you can see change the count
Known cards cannot be in an opponent's hand. For 54s, the table lists the combos left after one card of each rank is out of play.
| Known card | 54s combos left |
|---|---|
| Ace | 4 |
| King | 4 |
| Queen | 4 |
| Jack | 4 |
| Ten | 4 |
| Nine | 4 |
| Eight | 4 |
| Seven | 4 |
| Six | 4 |
| Five | 3 |
| Four | 3 |
| Three | 4 |
| Two | 4 |
A five or a four you can see leaves 3 of 4, so 25% of the combos are gone. Any card of a different rank leaves all 4 combos in play.
When the board is paired
A pair on the board uses up two of the four cards of its rank. The 54s combos that survive are in the table, one row per rank.
| Board pair | 54s combos left |
|---|---|
| Aces | 4 |
| Kings | 4 |
| Queens | 4 |
| Jacks | 4 |
| Tens | 4 |
| Nines | 4 |
| Eights | 4 |
| Sevens | 4 |
| Sixes | 4 |
| Fives | 2 |
| Fours | 2 |
| Threes | 4 |
| Twos | 4 |
Pairing fives or fours on the board cuts 54s to 2 combos. Every other paired board leaves the full 4.
A worked example
Take a hand of a five and a two. Your opponent then holds one of 1,225 possible hands, and 3 of them are 54s. That is 0.24%, or 1 in 408, instead of the 0.33% (1 in 306) you get when your cards do not block it.
How strong is 54s?
Rank 136 of 169 belongs to 54s, below the middle of the ranking, on 41.45% equity against a random hand. Equity here is wins plus half of the ties across all boards.
Add 54s to every hand ranked above it and you hold 1,026 of the 1,326 two-card hands, 77.38% of all deals.
54s against ranges of hands
A range is a list of hands, and equity against it weights each hand by its live combos. For 54s the five ranges give 25.13% against the top 5 percent (60 combos), 32.33% against the top 10 percent (130 combos), 35.80% against the top 20 percent (264 combos), 36.87% against the top 30 percent (386 combos) and 38.52% against the top 50 percent (658 combos).
| Opponent range | Combos in range | 54s equity |
|---|---|---|
| Top 5 percent | 60 | 25.13% |
| Top 10 percent | 130 | 32.33% |
| Top 20 percent | 264 | 35.80% |
| Top 30 percent | 386 | 36.87% |
| Top 50 percent | 658 | 38.52% |
54s is behind all five of these ranges, even the top 50 percent, so it is the underdog on average against each of them.
Holding 54s removes some premium combos from an opponent's reach: of the 130 in the top 10 percent, an average of 130.00 remain. That is no loss at all: 54s blocks none of those hands.
Hands next to 54s
A one-place difference in rank can be a large or small gap in a direct matchup. 54s has 47.86% equity against T2o, the hand ranked number 135: it wins 46.83% of boards, ties 2.04% and loses 51.12%. It is the underdog in that matchup. 54s has 37.28% equity against 85o, the hand ranked number 137: it wins 30.40% of boards, ties 13.76% and loses 55.84%. It is the underdog in that matchup.
Same ranks, other suits: 54o is the offsuit version, number 153 of 169 at 38.16%. 54o is 3 times as many combos but 3.30 points weaker against a random hand.
Questions about 54s combos
- How many combos of 54s are there?
- 4 combos, which is 0.30% of the 1,326 two-card starting hands, so you are dealt 54s about 1 time in 331.5.
- How many 54s combos are left if I hold a five?
- 3 combos of 54s remain out of 4. With a two as your other card, an opponent has it about 1 time in 408.
- Is 54s stronger than 54o?
- Yes, by 3.30 points: 54s has 41.45% equity against a random hand and ranks number 136, while 54o has 38.16% and ranks number 153. 54o has 12 combos to 54s's 4.
How this was computed
All 1,326 two-card holdings were grouped into classes, and the 54s group was counted. The blocker rows drop each combo containing a visible card; the board-pair rows do it for a visible pair.
Exact matchups, counted over all 1,712,304 boards, supply the rank and equity. A tie counts as half, and ranges are weighted by the combos that remain.
Source: GeorgeLiPoker's own count of the 52-card deck and its preflop matchup table.