A poker hand consist of 5 cards?A poker hand consist of 5 cards?Find the total number of possible five card poker card hands. Find the number of ways in which one can king can be selected?. Find...

A poker hand consist of 5 cards?

A poker hand consist of 5 cards?Find the total number of possible five card poker card hands. Find the number of ways in which one can king can be selected?. Find the number of ways in which four aces can be selected .Find the number of ways of getting four aces and one king. 

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lemjay | High School Teacher | (Level 3) Senior Educator

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In this problem, use the fundamental counting principle in which it states that if there are k, m and n ways to do each thing , then the number of ways in  doing all those things is `k*m*n` .

(A.) Find the total number of possible five card poker card hands.

Take note that in a poker, there are 52 cards. Also, the number of ways to select the cards is decreasing from the first to the fifth, since the selected card will no longer be available in the deck of cards to choose from. So, 

# of possible ways to select the 1st card : 52

# of possible ways to select the 2nd card : 51

# of possible ways to select the 3rd card : 50

# of possible ways to select the 4th card : 49

# of possible ways to select the 5th card : 48

------------------------------------------------------------------------------             Total # of ways to select poker card hands : `52*51*50*49*48 = 311875200 `      

Hence, the total numbers of ways to select poker card hands is 311,875,200 .

(B) Find the number of ways in which one king can be selected.

In a deck of cards, there are 4 kings. In order to select one king, we are going to assume that the first card is a king. Then, there are no more available king cards to choose from for the 2nd, 3rd, 4th and 5th card.

1st card - # of ways to select a king: 4

2nd card - # of ways to select a card that is not a king : 48

3rd card - # of ways to select a card that is not a king : 47

4th card - # of ways to select a card that is not a king : 46

5th card - # of ways to select a card that is not a king : 45

------------------------------------------------------------------------------             Total # of ways to select a poker card hands in which one is a king : `4*48*47*46*45 = 18679680`    

Thus, the number of ways to select a poker card hands in which one is a king is 18,679,680.

(C) Find the number of ways in which four aces can be selected.

There are 4 aces in a deck of cards. Let's assume that, we are going to have aces on the 1st, 2nd, 3rd and 4th card. Also,  for the 5th card, there are no more available aces on the deck of cards.

1st card - # of ways to select an ace :  4

2nd card - # of ways to select an ace : 3

3rd card - # of ways to select an ace : 2

4th card - # of ways to select an ace : 1

5th card - # of ways to select a card that is not an ace: 48

------------------------------------------------------------------------------             Total # of ways to select 4 aces in a poker card hands:` 4*3*2*1*48 = 1152`

Hence, the number of ways to have four aces is 1,152.


(D) Find the number of ways of getting four aces and one king.

Let's assume that we are going to have aces on the 1st, 2nd, 3rd, and 4th card. And, we will have a king on the 5th card.

1st card -  # of ways to select an ace : 4 

2nd card - # of ways to select an ace : 3

3rd card - # of ways to select an ace : 2

4th card - # of ways to select an ace : 1

5th card - # of ways to select a king : 4

-----------------------------------------------------------------------------             Total # of ways to select four aces and one king in a poker card hands. : `4*3*2*1*4 = 96`

So, there are 96 ways to select four aces and one king.

Sources:

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