Counting 5-card hands from a deck of standard playing cards. a 5-card hand is drawn from a deck of standard playing cards. how many 5-card hands have at least one club? (b) hown -card hands have at least two cards with the same rank?

5-card hands with at least one club:

5-card hands with at least two cards of the same rank:

Step-by-step explanation:

To determine how many 5-card hands have at least one club, we can count how many do NOT have at least one club, and then subtract that from the total amount of 5-card hands that there are.

A 5-card hand that doesn't have at least one club, is one whose 5 cards are from spades,hearts or diamonds. Since a standard deck of cards has 13 clubs, 39 of the cards are spades, hearts of diamonds. Getting a 5-card hand out of those cards, is choosing 5 cards out of those 39 cards. So there are 5-card hands without any clubs.

The total amount of 5-card hands is , since a 5-card hand is simply a group of 5 cards out of the full deck, which has 52 cards.

Therefore the number of 5-card hands that have at least one club is .

To determine how many 5-card hands have at least two cards with the same rank we can follow the same approach. We determine how many 5-card hands have NO cards with the same rank, and the subtract that out of the total amount of 5-card hands.

A 5-card hand that doesn't have cards of the same rank, is a group of 5 cards all from different ranks. Such hand can be made then by choosing first which 5 different ranks are going to be present in the hand, out of the 13 available ranks. So there are possible combinations of ranks. Then, choosing which card from each of the chosen ranks is the one that is going to be in the hand, is choosing which of 4 cards from EACH rank is going to be in the hand. So for each rank there are 4 availble choices, and so there are possible ways to choose the specific cards from each rank that will be in the hand. So the amount of 5-card hands with all ranks different is

Therefore the amount of 5-card hands with at least two cards with the same rank is  

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