Combinations

Stated simply, the number of combinations for a particular hand is the number of ways this hand can be dealt from a standard pack of 52 cards. In the case of the Royal Flush, this is four - since there is one Royal Flush that can be dealt for each suit in the deck.

J
♠

♠

♠

Q
♠

♠

♠

K
♠

♠

♠

For the Straight Flush, this number of combinations is thirty-six. This because there are 4 suits, and within each suit, there are 9 ways of getting a straight flush. 1-5, 2-6, 3-7, 4-8, 5-9, 6-10, 7-J, 8-Q, 9-K. The 10th in the list would be 10-A, which we have already seen is a Royal Flush, and is therefore omitted.

Probability

The probability is the likelyhood of getting dealt these 5 cards in any one deal. A probablity of 1 would mean a 100 percent chance of getting dealt a particular hand, and 0, well no chance. It is calculated by dividing the number of combinations for a particular hand, by the total number of five card poker hands. This number has been calculated to be 2,598,960.

To understand how we get this number, we look at the process of picking 5 cards from a total of 52.

The first card we pick has a total of 52 possible values, the next, 51 , and the next, 50 possible values and so on. Therefore, the number of different ways we can pick 5 cards from 52 is 52*51*50*49*48 = 311,875,200. We do however, still have one more step in our calculation.

Permutations

The number 52*51*50*49*48 is actually the number of ways you can pick 5 cards from a pack in a particular order - this is actually called a permutation.

10 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ J ♠ ♠ ♠ Q ♠ ♠ ♠ K ♠ ♠ ♠ A ♠ ♠

K ♠ ♠ ♠ A ♠ ♠ 10 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ Q ♠ ♠ ♠ J ♠ ♠ ♠

A ♠ ♠ 10 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ Q ♠ ♠ ♠ K ♠ ♠ ♠ J ♠ ♠ ♠

Q ♠ ♠ ♠ 10 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ K ♠ ♠ ♠ A ♠ ♠ J ♠ ♠ ♠

Four different permutations of the Royal Flush

When you have been dealt 5 cards, you are not usually concerned with the order. Since there are 5*4*3*2*1 (120) different ways of arranging our 5 cards (or combinations) we divide our previous calculation by this number to get the magic figure of 2,598,960. We are quite literally taking the order of the cards "out of the equation".

Summary

We can finally see that the total number of five card poker hands is b (52*51*50*49*48) / (5*4*3*2*1) = 311,875,200 / 120 = 2,598,960

In the table below, each type of poker hand is shown, along with the number of combinations, and the hand's probability. We have covered the first two probabilities. To learn how the other hand probabilities are calculated, read this article on 5-Card Poker Hands