Poker Hand Combination Calculator
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Introduction
Derivations for Five Card Stud
How to Calculate Possible Combinations of a Pocket Pair Hand If you randomly choose two cards from a deck, there are six different ways to make a single pocket pair. If you examine the possible ways you can be dealt 55, for example, you come up with 5 5, 5 5, 5 5, 5 5, 5 5, and 5 5. The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability. The next section shows how to derive the number of combinations of each poker hand in five.
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- A poker hand has five guards, so there are combination for taken four ways off getting the four jacks, and there would be a combination 48 taken. One wastes off, choosing the other card. We use 48 because a standard deck has 52 cards and four of them the four jacks had already been used, So we can only choose the one undercard from the 48.
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I have been asked so many times how I derived the probabilities of drawing each poker hand that I have created this section to explain the calculation. This assumes some level mathematical proficiency; anyone comfortable with high school math should be able to work through this explanation. The skills used here can be applied to a wide range of probability problems.
The Factorial Function
If you already know about the factorial function you can skip ahead. If you think 5! means to yell the number five then keep reading.
The instructions for your living room couch will probably recommend that you rearrange the cushions on a regular basis. Let's assume your couch has four cushions. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cushion, then there will be 3 positions left to put the second, 2 positions for the third, and only 1 for the last one, or 4*3*2*1 = 24. If you had n cushions there would be n*(n-1)*(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the fact(x) function in Excel will give the factorial of x. The total number of ways to arrange 52 cards would be 52! = 8.065818 * 1067.
The Combinatorial Function
Assume you want to form a committee of 4 people out of a pool of 10 people in your office. How many different combinations of people are there to choose from? The answer is 10!/(4!*(10-4)!) = 210. The general case is if you have to form a committee of y people out of a pool of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 people in your office in order. You could consider the first four as the committee and the other six as the lucky ones. However you don't have to establish an order of the people in the committee or those who aren't in the committee. There are 4! = 24 ways to arrange the people in the committee and 6! = 720 ways to arrange the others. By dividing 10! by the product of 4! and 6! you will divide out the order of people in an out of the committee and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combin(x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.
Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combin(52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.
Poker Math
The next section shows how to derive the number of combinations of each poker hand in five card stud.
Royal Flush
There are four different ways to draw a royal flush (one for each suit).
Straight Flush
The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.
Four of a Kind
There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.
Full House
There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.
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Flush
There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.
Straight
The highest card in a straight can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 45 = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.
Three of a Kind
There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 42 = 54,912.
Two Pair
There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 62 * 44 = 123,552 ways to arrange a two pair.
One Pair
There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 43 = 1,098,240 ways to arrange a pair.
Nothing
First find the number of ways to choose five different ranks out of 13, which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, leaving you with 1277. Each card can be of 1 of 4 suits so there are 45=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.
Specific High Card
For example, let's find the probability of drawing a jack-high. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 10-9-8-7 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights. Here is a good site that also explains how to calculate poker probabilities.Five Card Draw — High Card Hands
| Hand | Combinations | Probability |
|---|---|---|
| Ace high | 502,860 | 0.19341583 |
| King high | 335,580 | 0.12912088 |
| Queen high | 213,180 | 0.08202512 |
| Jack high | 127,500 | 0.04905808 |
| 10 high | 70,380 | 0.02708006 |
| 9 high | 34,680 | 0.01334380 |
| 8 high | 14,280 | 0.00549451 |
| 7 high | 4,080 | 0.00156986 |
| Total | 1,302,540 | 0.501177394 |
Ace/King High
For the benefit of those interested in Caribbean Stud Poker I will calculate the probability of drawing ace high with a second highest card of a king. The other three cards must all be different and range in rank from queen to two. The number of ways to arrange 3 out of 11 ranks is (11:3) = 165. Subtract one for Q-J-10, which would form a straight, and you are left with 164 combinations. As above there 1020 ways to arrange the suits and avoid a flush. The final number of ways to arrange ace/king is 164*1020=167,280.Internal Links
For lots of other probabilities in poker, please see my section on Probabilities in Poker.
Written by:Michael Shackleford
Poker combinations are important. Poker, as we all should know by now, is a game of skill and luck.
Fate and the dealer decide what cards you start off with, and that can make the difference between winning and losing a game.
There are 52 cards in a deck, which means that you can get up to 169 different combinations for your starting cards.
The cards you are dealt at the start do not make or break your game, but they play a huge role in defining how you progress.
We look at the best poker combinations of cards which you should be looking out for if you want to win big in poker.
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Aces and Kings
Probably the best starting poker combinations you could game are a pair of Aces or Kings, irrespective of the suit.
The chances of someone getting a pair is exceptionally remote but if you do get such a pair always think about raising the pre-flop.
A word of caution though, these hands are tough to beat but are not unbeatable.
Also, if you are playing with a couple of Kings, there is the possibility of someone playing with a couple of aces, so always tread carefully.
poker combinations: Queens, Jacks and an Ace-King pair
The next highest possible poker combinations are also based on the highest cards.
Again getting a Queen pair or a jack pair is extremely rare but the chances of winning a hand significantly increase.
Of course, they can still be beaten if the right moves are made but this is a strong hand to play with.
Poker Hand Combination Calculator Present Value
The Ace-King pair is another strong hand to think about.
This combination works well and gives you a lot of opportunities to make strong hands.
This isn’t as strong as a starting pair of high cards, but it is a good bet for a win at the table.
Ace-Queen, Ace-Jack
The next best poker combinations to start a poker game with is the Ace-Queen pair.
This is similar to the Ace-King pair but has a few limitations since the Queen card is ranked lower than the King card.
The Ace-Jack combination ranks below the Ace-Queen pair.
All the above pairs are made in the same suit, except in the case of a pair of cards which are of similar value.
Always the need to have same suit cards trumps cards of the same value.
So a Jack-Ace of the same color will trump, a Jack-Ace of different colors.
While these cards will undoubtedly play a huge role in deciding who wins a game.
It isn’t the end of the world if you are not dealt the above pairs.
There are many other ways to ensure you end up winning the game and we look at some of the best combinations of hands possible in the game of poker.
You might need to be a little sly in hoping that you get these cards from the dealer, but if you get the following combinations, you are guaranteed to win the game or at least do well.
Winner winner, chicken dinner!
Playing your way through the game is not easy but having the following hands will undoubtedly ensure victory and a nice amount of money for you to win.
Some of these hands are guarantees to win while some would require you to do some bluffing.
So without further to do, here are the best poker combinations you could have in a game of poker.
The Royal Flush
Royalty is vital in the game of poker and having a ‘Royal Flush’ is the best possible poker combinations of cards you could have after all the cards have been given to you.
A Royal Flush consists of the Ace, King, Queen, Jack and 10 of the same suit.
Nothing can beat this hand in a game of poker.
Straight Flush
Almost the next best thing to the Royal Flush, having 5 cards which are in a consecutive rank, for, eg, 5,6,7,8 and 9 of diamonds.
In case two players have a straight flush, the one with the higher denomination of cards wins.
Four of a kind
As the name suggests, having 4 cards of the same rank, high-value cards of the same rank will beat cards of a lower value.
Full House
Having three cards of the same rank and two other cards of a different rank.
In case of a tie, the highest ranking three cards of a player win the game.
Flush
This happens when a player has five cards which are of the same color but are not in any specific order.
This can include high cards with low-value cards.
Three of a kind
Having three cards which are of the same rank/strength, and two cards which are of a different color or unrelated side cards.
Two pair
Having two pairs of cards which are of the same value; for example, having two 10’s and two 6’s in hand.
One pair
In this hand, two cards must have the same value; and the other three cards can be of any other denomination.
High card
Just having one high card in hand should win you the game. In case any of the above hands don’t take place.
So there you have it, the best starting poker hand combinations which can set you out for glory. And the best possible hand combinations at the end of the game before you go on to bet and win.
Remember, poker isn’t an easy game to pick up; there will always be a lot of people who try different tactics to get a slight edge.
While having a Royal Flush in your hand is a guaranteed win; the other options aren’t.
You need to keep your wits about it. And one way of doing it is by practicing and keeping yourself informed.
Also, you should use a few tools to aid your game like poker calculators and odds calculators. They might not help you out in the table games, but will help you make decisions faster and help solve tables better.