How many possible -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 cards are there?
Suppose we drew card from a shuffled standard deck. There would be possibilities for this card. Then, we draw another card from the same deck. After the first card is drawn, the deck only contains cards, so there are possibilities for the second card.
This means that there are ways to draw cards from a standard deck. Does this mean that the number of possible hands of cards is
As it turns out, we're leaving an important piece of information out of this. Each possible hand of cards contains different permutations.
In other words, there are always different ways to draw the same hand. Since we know that there are permutations for each hand, we can simply take and this is the number of possible hands of cards.
In mathematical terms, we've just computed the number of combinations of cards out of the in a standard deck. In mathematical notation, it looks like this:
The two different notations on the left mean the same thing, but the binomial coefficient is more common when dealing with poker probabilities.
The number of possible -card hands in poker will be expressed as and to calculate it, we'll follow a similar line of thinking. First, we'll compute the number of ways to draw cards from a deck.
There are possibilities for the first card, then possibilities for the second card, and so on. This means there are ways to draw cards from the deck.
Then, we need to account for all the permutations of the same hand of cards. Fortunately, this is a very similar calculation: There are possible ways to choose the first card, then ways to choose the second card, and so on. In total, there are permutations of cards.
Each hand has the same number of permutations, so the number of possible -card poker hands is
This is a staggering amount. If you stacked the possible -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 cards of the same suit when drawing cards from a shuffled deck.
There are possible suits, and each suit has cards. To form this type of hand, we want of those cards, so the probability is