As you can probably tell, school has been keeping me busy. I've been leaning about counting. Sounds easy, right? Well, counting is a type of discrete math. We have learned to count the number of different ways we can arrive at a desired outcome based upon certain conditions. If you've ever taken a statistics class you probably started off by working with combinations and permutations. This is the type of stuff we're working on.
One of our recent assignments involved counting the number of different five-card poker hands. For instance, there are 2,598,960 different hands that you can have. We get this number by considering how many possibilities you have for every card you draw. There 52 different cards you could get for your first card. For your next card, you now only have 51 different choices. So for all five cards, you have (52)(51)(50)(49)(48) possibilities. Actually, that's not quite true; that product would give you the total possibilities if the order in which you got the cards mattered. Since that's not the case, you have to divide by the total number of ways to order five cards (because each possibility was counted this many times). This comes out to 5! ("five factorial") which is the same as (5)(4)(3)(2)(1). This is how we know there are over 2.5 different poker hands. Then we counted how many named hands there were (ie, royal flush, full house, three of a kind, etc). It probably comes a no surprise that the order in which hands beat other hands directly corresponds to how many different ways there are to get those hands. Here are the final counts we came up with: royal flush - 4; straight flush - 36; four of a kind - 624; full house - 3,744; flush - 5,108; straight - 10,200; three of a kind - 54,912; two pair - 123,552; one pair - 1,098,240; and "nothing" - 1,302,540.
Posted by Matthew at May 28, 2005 10:13 PM