By now, hopefully you’ve had a chance to read some or all of my numerous posts on calculating hand odds for Texas Hold’em. Let’s have a look at some more advanced calculations, including determining the number of possible specific hands. (Warning: math ahead. But it’s easy, I swear.) With cards, order of dealing sort of matters. As far as betting goes, the order cards appear makes a difference to your betting. Obviously, an A-A in the pocket is more beneficial than A-x followed by an A in the flop. But as far as determining the value of your hand, the order of the 5 cards you get (2 in the pocket, 3 from any of the five community cards) doesn’t matter. So how do we factor this in?
First, we need some notation. In the mathematical field of game theory, two important terms are used: combinations and permutations. They are both from the mathematical field of combinatorics and optimization, which is a subset of statistics, and a superset of game theory.
A combination of objects doesn’t care about the order in which they are picked. A permutation does. For example, the number 967 is made of the digits 9, 6, and 7. So is 796, but it is a different permutation than 967, because order matters.
To calculate the number combinations or permutations related to a poker hand, we first need to understand the factorial notation. Don’t let the math terms scare you. It’s actually a really simple concept:
Factorial notation: n! = n x (n-1) x (n-2) x … x 1
0! = 1 [Don’t worry; it’s just the definition]
1! = 1
2! = 2×1 = 2
3! = 3×2×1 = 6
4! = 4×3×2×1 = 24
Pretty easy. …