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How many possible 55-card hands of poker are there?

The simple answer to this question is — there are a lot. But to give a precise answer to this question, let's start with something much less complicated: how many possible hands of 22 cards are there?

Suppose we drew 11 card from a shuffled standard deck. There would be 5252 possibilities for this card. Then, we draw another card from the same deck. After the first card is drawn, the deck only contains 5151 cards, so there are 5151 possibilities for the second card.

This means that there are 52×51=265252 \times 51=2652 ways to draw 22 cards from a standard deck. Does this mean that the number of possible hands of 22 cards is 2652?2652?

As it turns out, we're leaving an important piece of information out of this. Each possible hand of 22 cards contains 22 different permutations.

You can draw these cards in two different orders, but they both represent the same hand of two cards. You can draw these cards in two different orders, but they both represent the same hand of two cards.

In other words, there are always 22 different ways to draw the same hand. Since we know that there are 22 permutations for each hand, we can simply take 2652÷2=1326,2652 \div 2 = 1326, and this is the number of possible hands of 22 cards.

In mathematical terms, we've just computed the number of combinations of 22 cards out of the 5252 in a standard deck. In mathematical notation, it looks like this:

52C2=(522)=1326._{52}\text{C}_{2}=\binom{52}{2} = 1326.

The two different notations on the left mean the same thing, but the binomial coefficient ()\binom{\cdot}{\cdot} is more common when dealing with poker probabilities.

The number of possible 55-card hands in poker will be expressed as (525),\binom{52}{5}, and to calculate it, we'll follow a similar line of thinking. First, we'll compute the number of ways to draw 55 cards from a deck.

There are 5252 possibilities for the first card, then 5151 possibilities for the second card, and so on. This means there are 52×51×50×49×4852 \times 51 \times 50 \times 49 \times 48 ways to draw 55 cards from the deck.

Then, we need to account for all the permutations of the same hand of 55 cards. Fortunately, this is a very similar calculation: There are 55 possible ways to choose the first card, then 44 ways to choose the second card, and so on. In total, there are 5×4×3×2×15 \times 4 \times 3 \times 2 \times 1 permutations of 55 cards.

Each hand has the same number of permutations, so the number of possible 55-card poker hands is

(525)=52×51×50×49×485×4×3×2×1=2598960.\begin{aligned} \binom{52}{5} &= \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} \\ &= 2598960. \end{aligned}

This is a staggering amount. If you stacked the possible 55-card hands on top of one another, the stack would tower over the tallest building in the world (the Burj Khalifa in Dubai) and would rival the height of mountains.

You can use the same method above to compute the number of specific types of hands in poker, and then find the probability of those hands. For example, let's suppose we wanted to find the probability of drawing 55 cards of the same suit when drawing 55 cards from a shuffled deck.

There are 44 possible suits, and each suit has 1313 cards. To form this type of hand, we want 55 of those cards, so the probability is

4×(135)(525)=514825989600.00198.\begin{aligned} \frac{4\times \binom{13}{5}}{\binom{52}{5}} &= \frac{5148}{2598960} \\ &\approx 0.00198. \end{aligned}

Today's Challenge

The Jack, Queen, and King are the face cards of a standard poker deck.

This is an example of the type of hand described below. This is an example of the type of hand described below.

If you draw 55 cards from a shuffled poker deck, what is the approximate probability that all 55 cards are face cards?

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