# Some Mathematical Poker Questions (Discrete Math)

Mathematically speaking, I have two questions:

1. How many ways can a 5-card draw-poker hand be dealt?
2. How many 5-card draw-poker hands exist?

Thanks!

## 3 Answers

I think your first question is asking, "How many ways can one be dealt 5 consecutive cards from a standard deck of 52 cards with the order of cards being distinct hands?" That is to say, a hand with the first card is Ace-of-spades then 2-of-spades then 3 other cards is distinct from being dealt first the 2-of-spades, next the Ace-of-spades, then the 3 other cards. In this case, you are looking for a permutation of the number of ways to order 5 cards from a set of 52 objects. Alternatively, this is asking for the number of ways to leave behind 47 (52-5) cards in a particular order from the deck box. So the formula for a permutation of k items out of n items [notation for a Permutation is n_P_k]is n!/(n-k)! (Look up the Factorial Function) to understand what n! means. To answer your question, we have n=52 cards in the deck, and k=5 cards per hand. So the answer is 52_P_5 = 52!/47! = 52*51*50*49*48 = 311875200.

Your second question, "How many 5-card poker hands exist?", is equivalent to asking how many ways to combine 5 cards together from a deck of 52. Note how this is different than the above permutations because in a Combination, order does not matter. So the formula (some like to say "n-choose-k") can be expressed with the Binomial Coefficient which is hard to demonstrate in this answer box, but it uses parentheses and almost looks like a fraction. Sometimes it is written n_C_k. The formula is n! / [k!(n-k)!]. You might also want to think of it as a permutation divided by k! to avoid repetition when it comes to the order of the items, so you could even say n_C_k = (n_P_k)/k! In your example, n=52 and k=5 like before so we calculate in two ways:

52_C_5 = (52!)/[5!*47!] = 2598960

52_C_5 = (52_P_5)/5! = 311875200/5! = 311875200/120 = 2598960

To calculate all the 5 Card draw poker hands that exist you have to calculate

51! / 47!

or explicite way 52 x 51 x 50 x 49 x 48 = 311,875,200

Now you need to know that for poker it doesn't matter wheter you have spades or clubs or whatever color. So in terms of value there are less hands. But that's the number of different hands you can have.

What do you mean by "how many ways can a 5-card draw-poker hand be dealt ?

like 5 cards to one person or first 1 than 2 than 3 cards etc?

as soon as I know, i calculate that.

How many ways can a 5-card draw-poker hand be dealt?

In order to calculate the number of ways that a hand can be dealt, meaning that the order of the cards is significant, you need to multiply the possible number of cards at each step together.

For a particular hand (meaning 5 specific cards), when the first card is dealt, you have 5 possible cards that would "qualify". When the second card is dealt, we assume that the first one has already been dealt, so there are now 4 possible cards. When the third card is dealt, there are 3 cards left in the deck. Continue this until there is only one possibility as the fifth card is dealt. The math would look like this: 5*4*3*2*1 = 120 different ways when you consider all of the possible different orders.

In a similar manner, for a particular "group" of poker hands (meaning flush, straight, pair, etc) you would need to adjust the number of cards accordingly. For instance, the number of ways to deal a flush would be: 52*12*11*10*9 = 617,760 (note that this would include straight flushes as well, so to have a "true" answer on how many ways are there to deal a flush, we would need to calculate the same for straight/royal flushes and subtract them from this answer).

How many 5-card draw-poker hands exist?

Now, a poker hand doesn't care about the order of the cards or the suit (unless they are all the same suit in which case the actual suit itself doesn't matter, just the fact that they are the same). The total possible number of 5-card poker hands is 7,462 with 2,598,960 different variations (for instance, there is one possible hand in the royal flush group (T-J-Q-K-A, all in the same suit), but four different variations (suits) that it can be dealt in.)

Wikipedia also has a good explanation of the poker probabilities.