This page provides proof that there are 22,100 possible flops in poker.

According to the rules of poker, the **order of the cards on the flop is irrelevant**, so 3c 4d Kh is the same as 4d Kh 3c. Only the number of **different poker flops** will be considered.

There are 52 poker cards. 13 different card ranks, and 4 different suits.

Card Ranks: 2 3 4 5 6 7 8 9 Ten Jack Queen King Ace

Suits: Clubs Diamonds Hearts Spades.**13 x 4 = 52**

We can simplify the process by designating each different poker card with a number, and **let the numbers 1 to 52 each represent a unique poker card**.

**Let the set of numbers {1,2,3, ... 50,51,52} represent all of the cards in poker.**

Using combination mathematics, all possible combinations of poker cards in a flop will be**C(n,k)** where n is the number of poker cards (elements in the set of all poker cards), and k is the number of cards in the combination (representing the flop)**C(52,3)**

The number of k-combinations from the set of all poker cards with n (52) elements is the binomial coefficient (also known as the 'choose function'):**C(n,k)** is defined by the formula**___n!___k!(n - k)!**

Substituting n and k we have

3!(52 - 3)!

A simple and easily understandable way to determine the number of unique flops without any doubt is to count them.

We can start counting flops like this:

1 2 3 (notice that each card is different)

1 2 4

1 2 5

...

1 2 52

1 3 4 (notice here that we incremented the middle card, but the right card did not start at 2 again because 1 3 2 would be the same as 1 2 3, which we have already counted!)

So essentially we can create a formula for generating unique flops. **Lets call the Cards in a flop: a, b and c**.

Since the order of the cards on the flop don't matter: 4,13,51 == 13,51,4 etc, any given flop is determined by any one permutation [ (a,b,c), (c,b,a), etc ]

So any flop can be associated with (a,b,c) where a < b < c.

And every unique flop can be represented as (a,b,c) where a < b < c.

The formula: **a = 1 to 52, b = (a + 1) to 52, c = (b + 1) to 52** will generate every possible unique flop in the game of poker.

This algorithm will hit all ordered triples **(a,b,c)** where **1 <= a < b < c <= 52**

Luckily this is the 21st Century and we have computers which can perform 'zillions' of calculations in a second.

Conveniently Visual Basic is a simple understandable lanuage which has syntax similar to the formula we created above.

Dim a As Long, b As Long, c As Long, Count As Long

___For a = 1 To 52

______For b = a + 1 To 52

_________For c = b + 1 To 52

____________Count = Count + 1

_________Next c

______Next b

___Next a

MsgBox "There are " + Count + " possible poker flops!"

The code produces: **There are 22100 possible poker flops!**

Thanks to DaMancha on EfNet #math for the tips on mathematical writing.