Linda Mae swerved and headed my way after she saw me sign up for a few games. She went to Atlantic City over the weekend and someone there told her there were only 7,462 poker hands. "Was that right?" she demanded, "because you always say there are over two-and-a-half million poker hands!"
"We're both right," I answered. "There are 2,598,960 possible five-card hands from a standard deck. In poker games the particular suits don't matter, so there are 7,462 different hands. Let's do the math." She first rolled her eyes and then agreed to pay attention.
"There are 156 different fours-of-a-kind: four aces with a king kicker differs from four aces with a queen kicker, and so on. There are 13 ranks for the quads, times 12 ranks for the kicker equals 156. Counting suits there are 624 different fours-of-a-kind possible. For the same reason there are 156 types of full houses. Counting suits, there are 3,744 different full houses possible.
"Why didn't you start with straight flushes," she asked.
"Straights, flushes and straight flushes are special cases of unpaired hands," I answered, "Don't worry, we won't forget them.
"Continuing on, there are 858 different types of three-of-a-kind, as determined by 13 ranks times C(12,2) for the other two cards. Counting suits, there are 54,912 different threes-of-a-kind. Similarly, there are 858 types of two-pair hands, as given by C(13,2)*12. Counting suits, there are 123,552 different two pair hands.
"There are 2,860 one-pair hands, from the expression 13*C(12,3), meaning any one of 13 ranks for the pair and any three of the 12 remaining ranks for the other cards. Counting suits, there are a whopping 1,098,240 one-pair hands possible.
"From the expression C(13,5) we find there are 1,287 types of hands without any duplicated ranks. Now it gets kind a' tricky," I told her, "Ten of those hands have no-gap sequences from A-KQ- J-T to 5-4-3-2-A, so there are 1,277 hands without a sequence. There are 45, or 1,024 possible combinations of suits, of which 4 are all the same suits, leaving 1,020 non-flush suit combinations. There are ten types of straight flushes and ten types of straights; counting suits there are 40 straight flushes and 10,200 straights. Continuing in a like manner, there are 5,108 flushes (4*1,277) and 1,302,540 nothing hands (1,020*1,277). And now we're done."
"Whoosh," said she, "What a lot of work. Does all that add up to 7,462?"
"Not all that much work," I responded, "The table shows our results."
"So you were both right!" she declared. "Since there are so few types of hands compared to possible hands, why do you purposely confuse us by using millions of hands in your columns?"
"We use all the hands possible when we're concerned with probabilities. For example, look at the straight flush and straight rows. While there are ten types of each, the possible straights exceed the possible straight flushes by 254-to-1. In other words, a straight flush is 255 times less likely than a straight. We don't ever use types of hands to compute probabilities and odds.
"Of course on-line poker sites and poker room dealers always use types of hands to determine who won what."
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net
