Mathematics, 13.09.2019 17:30 georgesarkes12

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?

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Answer from: kyandrewilliams1

5-card hands with at least one club: ${52 \choose 5}-{39 ]choose 5}$

5-card hands with at least two cards of the same rank: ${52 \choose 5}-{13 \choose 5}4^5$

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 ${39 \choose 5}$ 5-card hands without any clubs.

The total amount of 5-card hands is ${52 \choose 5}$ , 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 ${52 \choose 5}-{39 \choose 5}$ .

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 ${13 \choose 5}$ 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 4^5 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 ${13 \choose 5}\cdot{4^5}$

Therefore the amount of 5-card hands with at least two cards with the same rank is ${52 \choose 5}-{13 \choose 5}\cdot4^5$

Answer from: Quest

Cwill be your correct answer
Asap 30 points the amount of money sheryl makes in a day is directly proportional to the amount of h

Answer from: Quest

16.9 times 2.85

step-by-step explanation:

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