How many possible flops?

During Tuesday night's WSOP broadcast, just before a commercial break, ESPN flashed on the screen this "Poker Fact": "There are 19,600 possible flops in Texas Hold'em."

I wonder how many people checked them on that.

I did. Yeah, that's how pedantic I can be. (This is the point as which the haters click on "submit a comment" in order to make a wisecrack about obsessive-compulsive disorder.)

The answer is not correct. Or, at least, it's not clearly correct, and it's not the answer I would have given.

The number of combinations of cards you can select is given by a straightforward (though often cumbersome-to-calculate) formula: see here for details. Fortunately, spreadsheets have a built-in function for this, reducing the work to a fraction of a second. That makes it easy to determine that C(52,3) (i.e., the number of combinations of three cards from a 52-card deck) is 22,100.

So how did ESPN come up with 19,600? Apparently they are assuming a 50-card deck, because C(50,3) is indeed 19,600. In other words, they are providing the answer to a slightly different question than the one they were asking.

Before the dealer shuffles the cards for a hand, if you ask how many different flops might theoretically come up in the next hand, you would have to say 22,100. Of course, if you are a player in the game and know your own two cards, then the universe of possible cards is reduced to 50. You might then say that the number of possible flops is 19,600, because you can eliminate the 2500 flops that contain one or both of your hole cards. But then you are not answering the question "How many possible flops are there in Texas Hold'em?" but, rather, "How many possible flops are there in Texas Hold'em, given that your two specific hole cards are known to be unavailable?"

Even that is a little bit dicey, because once the dealer has shuffled and cut the deck, there is only one possible flop that can come (barring dealer error). So if you're asking the question after the deal, as is implied in the answer that ESPN gave, the answer might better be 1 than 19,600.

The 19,600 is usually going to be the more useful number when you are doing post-hoc and/or theoretical analysis of a hand. But I submit that ESPN got it wrong. When asking the number of "possible flops," without specifying any preconditions or limitations or exclusions, the answer has to be 22,100.