Proof that there are 22,100 possible poker flops

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

Unique poker flops and the order of the cards

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.

Number of Poker Cards, Ranks and Suits

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

Proof for the number of possible poker flops

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.

Mathematical Proof for the number of poker flops

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)
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
k!(n - k)!

Substituting n and k we have
3!(52 - 3)!

= 22,100 possible poker flops.

Algorithmic Proof for the number of poker flops

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.

Simple code to count the number of poker flops

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.

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