# 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