 # Probability of Royal Flush

→

Calculate the probability of a Royal Flush:                    First calculate the total number of possible hands in a 52 card deck:
From a deck of 52 cards, we want the number of possible unique ways we can choose 5 cards.
Using the combinations formula 52 choose 5 shown here, we get:
 Total Possible 5 Card Hands  = 52! (52-5)! * 5!

 Total Possible 5 Card Hands  = 52! 47! * 5!

 Total Possible 5 Card Hands  = (52 * 51 * 50 * 49 * 48) * 47! 47! * (5 * 4 * 3 * 2 * 1)

## Cancelling the 47! on top and bottom we get:

 Total Possible 5 Card Hands  = 311,875,200 120

Total Possible 5 Card Hands = 2,598,960

Possible Royal Flushes = Possible ways to get 10,J,Q,K,A all suited
Possible Royal Flushes = 1 way * 4 possible suits (Spade,Heart,Diamond,Club) = 4
 Probability(Royal Flush)  = Possible Royal Flushes Total Possible 5 Card Hands

 Probability(Royal Flush)  = 4 2,598,960

Using our GCF Calculator, we see that 4 and 2598960 can be reduced by 4
Reducing top and bottom by 4, we get:
 Probability(Royal Flush)  = 1 649,740

In decimal format, this probability is equal to approximately 1.5391E-6

In decimal format, this probability is equal to approximately 1.5391E-6

### How does the 5 Card Poker Hand Calculator work?

Calculates and details probabilities of the 10 different types of poker hands given 1 player and 1 deck of cards.
This calculator has 1 input.

### What 1 formula is used for the 5 Card Poker Hand Calculator?

1. Total Possible 5 Card Hands = 2,598,960

For more math formulas, check out our Formula Dossier

### What 4 concepts are covered in the 5 Card Poker Hand Calculator?

5 card poker hand
combination
a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter
nPr = n!/r!(n - r)!
factorial
The product of an integer and all the integers below it
probability
the likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes