Math-Prof HOME Probability Table of Contents Ask A Question PREV  

More Shortcuts
Defining Combinations

 

 There are 52 cards in a regular deck.

 Say you are being dealt a poker hand.
 The first card you get can be any one of the 52 cards.
 That means the possible number of first cards
 that you could get is ...
 

 52

 
 After you get your first card, there are still 51 cards out there.
 Other players will get some of them.
 But since we don't know which ones they get,
 it works like all of the cards are just out there 
 in the same unknown zone.
 As far as we know or can tell,
 we might get any one of them
 
 That means the number of combinations 
 of first and second cards that we might get is ...
 

52 51 = 2,652

 
 There are also ...
 
 50 possible 3rd cards you could get
 49 possible 4th cards you could get
 48 possible 5th cards you could get
 
 That makes the number of different ways
 that you can be dealt a 5 card poker hand is ...
 

 52 51 50 49 48

 

 That's all true and wonderful, 
 but math types like to write stuff in as small a space as they can. 
 Here's what they do this time ...
 
 There is a rule in math that says ...

 

 
 If you write the rule backwards, you get ...
 

 

 So what?

 How does that help us with ...
 

 52 51 50 49 48

 
 This is not a real factorial,
 it would also need to have the numbers from 47 down to 1 
 multiplied times the numbers we do have.

 All of those numbers together make ...

 

 47!

 
 If we want to, we can write 52! as ...
 

52 51 50 49 48 47!

 
 So we can say ...
 

 
 and that can be written as ...
 

 
 Well that's just great ...
 Almost.
 
 We still have one more little problem.
 The way the numbers work.
 These hands, which all have the same cards ...
 

 
 would be counted as different hands.
 They all have the same cards in them, 
 but the cards were dealt in a different order.
 Most of the time (like now), 
 the order in which the cards are dealt
 does not matter.
 
 Just how many ways can you deal the same 5 cards?
 
 You can get any of the 5 as a first card.
 Any one of the remaining 4 as the second card.
 And so on. Putting these all together, we get ...
 

5 4 3 2 1 = 120

 
 Since most of the time we don't care about the card order.
 Our formula for the number of possible hands ...
 

 
 gives an answer that is way too big.
 Each possible hand shows up 5! times.
 To make the numbers work out right
 we need to divide the formula by 5!.
 An easy math way to do that is to put 5! in the denominator.
 That makes the whole formula ...
 

 
 What we have here is ...
 

 
 Example:
 
 How many different 7 card hands are there
 in a regular 52 card deck?
 (We don't care about the order of the cards)
 

 
 Simplifying this so we can multiply, we get ...
 

 
 Multiplying this out, we get ...
 

 
 Doing the division ...
 

 
 So there are just over 133 million different 7 card hands
 if we don't care what order we get the cards in.
 
 Pop Quiz:
 
 How many different 7 card hands are there
 in a 52 card deck if we DO care about the card order.
 
 
 Answer:
 Over 674 billion.
 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Probability Table of Contents Ask A Question PREV