 # Probability of Choose Your Hand

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## Calculate the probability of drawing a AKKQJ

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

## Calculate the probability of drawing Ace There are 4 A cards in the deck and 52 total cards in the deck to choose from
 Probability of drawing A  = 4 52

## Calculate the probability of drawing King There are 4 K cards in the deck and 51 total cards in the deck to choose from
 Probability of drawing K  = 4 51

## Calculate the probability of drawing King There are 3 K cards in the deck and 50 total cards in the deck to choose from
 Probability of drawing K  = 3 50

## Calculate the probability of drawing Queen There are 4 Q cards in the deck and 49 total cards in the deck to choose from
 Probability of drawing Q  = 4 49

## Calculate the probability of drawing Jack There are 4 J cards in the deck and 48 total cards in the deck to choose from
 Probability of drawing J  = 4 48

## Calculate final probability:

Since each card draw is independent, we multiply each of our 5 card draws
 P(AKKQJ)  = 4 x 4 x 3 x 4 x 4 52 x 51 x 50 x 49 x 48

 P(AKKQJ)  = 768 311875200

 Probability(Choose Your Hand)  = 768 311,875,200

Using our GCF Calculator, we see that 768 and 311875200 can be reduced by 384
Reducing top and bottom by 384, we get:
 Probability(Choose Your Hand)  = 2 812,175

In decimal format, this probability is equal to approximately 2.4625E-6