Suppose you have a standard deck of 52 playing cards. A deck of 52 cards consist

Question

Suppose you have a standard deck of 52 playing cards. A deck of 52 cards consists of 4 suits and within each suit there are 13 cards of different ranks: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. A hand is a selection of some number of cards from the deck.

The order of the cards within a hand is not important.

a. How many different five-card hands are there? (i.e. if you pick 5 cards form the deck, how many combinations are available?)

b. How many different five-card hands are there such that they contain cards with identical rank (e.g. all of them are Queen)?

c. How many different five-card hands are there such that they contain only Jack, Queen or King?

Explanation / Answer

A) total number of different five card hands = 52C5

= 52! / (5! * 47!)

= 2598960

B) number of different five card hands with identical rank = 0, because there 4 cards of each rank. So the fifth card has to be of a different rank.

C) number of different five card hands which contains only jack, queen or king = 12C5 = 12! / (5! * 7!) = 792

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