This question refers to a standard deck of playing cards. If you are unfamiliar with playing cards, there is an explanation in Probability of an event' section under the heading Standard playing cards. A five-card hand is just a subset of 5 cards from a deck of 52 cards. How many different five-card hands are there from a standard deck of 52 playing cards? How many five-card hands have exactly two hearts? How many five-card hands are made entirely of hearts and diamonds? How many five-card hands have four cards of the same rank? A full house" is a five-card hand that has two cards of the same rank and three cards of the same rank. For example, (queen of hearts, queen of spades, 8 of diamonds, 8 of spades, 8 of clubs). How many five-card hands contain a full house? How many five-card hands do not have any two cards of the same rank?

Number of ways = 2595960

P(two hearts) = 0.003 %

P(hearts and diamonds) = 63.73 %

P(4 same rank) = 0.024 %

P(full house) = 0.144 %

P(No same rank) = 50.7 %

Explanation:

How many different five-card hands are there from a standard deck of 52 playing cards?

There are total 52 cards out of which we have to select 5 cards

Number of ways = 52C5

Number of ways = 2595960

There are 2595960 different ways of dealing five-card hands

How many five-card hands have exactly two hearts?

There are total 13 hearts out of which we have to select two hearts

P(two hearts) = 13C2/52C5

P(two hearts) = 78/2598960

P(two hearts) = 0.003 %

How many five-card hands are made entirely of hearts and diamonds?

P(hearts and diamonds) = 13C5*13C5/52C5

P(hearts and diamonds) = 1287*1287/2598960

P(hearts and diamonds) = 1656369/2598960

P(hearts and diamonds) = 63.73 %

How many five-card hands have four cards of the same rank?

P(4 same rank) = 13C1*12C1*4C1/5C2

P(4 same rank) = 13*12*4/2598960

P(4 same rank) = 624/2598960

P(4 same rank) = 0.024 %

How many five-card hands contain a full house?

P(full house) = 13C2*2C1*4C3*4C2/5C2

P(full house) = 78*2*4*6/2598960

P(full house) = 3744/2598960

P(full house) = 0.144 %

How many five-card hands do not have any two cards of the same rank?

P(No same rank) = 13C5*4C1*4C1*4C1*4C1*4C1/5C2

P(No same rank) = 1287*4*4*4*4*4/2598960

P(No same rank) = 1317888/2598960

P(No same rank) = 50.7 %