Seven cards are selected from a standard deck of cards. a) Determine the probability that exactly 5 of them are hearts. b) Determine the probability that there are 3 hearts and 3 diamonds. c) Determine the probability that there are 3 hearts, 3 diamonds and 1 spade. d) Determine the probability that there are 2 Aces and 2 Kings. e) Determine the probability that there are 2 Aces and 3 Kings.

a

b

c

d

e

Step-by-step explanation:

From the question we are told that

   The number of cards selected is  n = 7

Generally in a standard deck of cards

   The total  number of cards is  N  =  52

    The number of  hearts is  h = 13  

    The  number of diamonds is  d = 13

    The number of  spade is  s  =  13

    The number of  Ace is  a = 4

    The number of kings is  k =  4

Considering question a

   The number of ways to selected 5 hearts out of the  13 hearts is mathematically represented as

Here C means combination so  

The number of cards that is not hearts  is  52 -  13  =  39

Generally the number of ways of selecting the reaming 2 cards is  

Generally the number of ways to select the 7  cards from the 52 deck of cards is  

Generally the probability that exactly 5 of the 7 cards  are hearts is mathematically represented as

=>  

=>  

Considering question b

     The number of ways to selected 3 diamonds out of the  13 diamonds is mathematically represented as

The number of ways to selected 3 hearts out of the  13 hearts is mathematically represented as

The number of cards that is not hearts or diamond  is  52 -  (13 +13)  =  26

Generally the number of ways of selecting the reaming 1 cards is  

Generally the probability that there are 3 hearts and 3 diamonds is

=>

=>  

Considering question c

The number of ways to selected 1 spade out of the  13 spade is mathematically represented as

Generally the probability that there are 3 hearts, 3 diamonds and 1 spade is

=>

=>  

Considering question d

The number of ways to selected 2 Aces out of the  4 Aces is mathematically represented a

The number of ways to selected 2 Kings out of the  4 Kings is mathematically represented a

The number of cards that is not Aces or Kings  is  52 -  (4 +4)  =  44

Generally the number of ways of selecting the reaming 3 cards is  

Generally the probability that there are 2 Aces and 2 Kings is

=>  

=>

Considering question e

The number of ways to selected 3 Kings out of the  4 Kings is mathematically represented a

Generally the number of ways of selecting the reaming 2 cards is  

Generally the probability that there are 2 Aces and 3 Kings is

=>  

=>  

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Step-by-step explanation:

Look at attached image for answer.