A One-Question Quiz

You probably have played hundreds of thousands, maybe even millions, of hands of hold'em. Yet, preflop, there is a relatively limited number of hands that one player can hold, and you certainly know what they are. Just in case you don't, I have the following quiz question (after you finish the quiz, I'll explain why it's important): How many possible two-card starting hands are there for any one player?

So, what was your answer? If you're like most players, you hemmed and hawed, and said that it is not very important to know exactly how many there are, but you have a rough idea. I'm going to make the question a little easier for you; I'll make it multiple-choice:

How many possible two-card starting hands are there for any one player?

a. 169
b. 1,326
c. Both a and b
d. Neither a nor b

Drumroll, please. And the answer is: c. I'm not kidding; both 169 and 1,326 are correct. There are 169 hands if you don't count equivalent hands separately. That means that, for example, A-K suited is counted as one hand. There are 1,326 starting hands if you do count equivalent hands separately. That means, for example, that A-K suited counts as four hands (A-K suited of spades, hearts, diamonds, and clubs).

Not counting equivalent hands separately is important in deciding your strategy for each hand in each situation. Thus, you really have to know how to play only 169 different hands, which consist of 13 pairs, 78 suited combinations, and 78 unsuited combinations. For practical purposes, you can follow the same strategy with groups of hands. For example, all ace-baby suited hands (A-5, A-4, A-3, and A-2 suited) probably can be handled the same way. I don't know any top player who makes a distinction between these hands strategically. That is pretty much the totality of the usefulness of 169. It is very misleading when you try to do any calculation of odds or percentages.

The number 1,326 can be derived in a number of ways. If we start from the 169 possible hands, we find that each pair can be made six ways. Each suited combination can occur four ways. Each unsuited combination can be made 12 ways. So, the total is 13 pairs times 6, or 78; 78 suited combinations times 4, or 312; and 78 unsuited combinations times 12, or 936. Another way to arrive at this number is to say that the first of your two cards can be any of 52 cards. The second can be any of the remaining 51. If we multiply 52 times 51, we get the number of two-card permutations (order counts; the A K is considered different than the K A). To get the number of combinations (order doesn't matter), we divide by 2. So, we get 52 times 51 divided by 2, which is 1,326.

In subsequent columns, I'll discuss different methods of using the number 1,326 to make calculations for various hands and situations, but let's take a simple example to end this column. Your opponent raises and you know that he has A-A, K-K, A-K suited, or A-K offsuit. How often will he have each of these four hands? (If you are using 169, you might think these four hands are equally likely.) The answer is A-A -- 6, K-K -- 6, A-K suited -- 4, and A-K offsuit -- 12, for a total of 28 ways out of 1,326, or, more precisely, 28 out of 1,326 if your two cards don't contain an ace or a king. Suppose that you have the A K; then what are the chances of each combination? A-A and K-K can now occur only three ways each. A-K suited can occur only two ways. A-K offsuit can occur seven ways. While these particular numbers may not be crucial, it is essential to understand how many possibilities there are, and how they are calculated. In my next column, I will discuss how the number 1,326 is used in creating an ordered list of hold'em hands by percentile.

Steve "Zee" Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. He can be found at many major tournaments and playing on Full Tilt, as one of its pros. When escaping from poker, he hangs out in his bars on Avenue A in New York City -- Nice Guy Eddie's on Houston and Doc Holliday's on 9th Street.