Question

in a certain card game , each player is initially dealt hand consisting of 5 cards from a deck of 52 cards. 52 deck cards consist of 4 suits and 13 rank in each suites. how many ways you can deal a hand that contain at least one card from every suit

Answer 1

To calculate the number of ways to deal a hand that contains at least one card from every suit in a certain card game, you can follow these steps: calculate the total number of possible hands, calculate the number of hands that do not contain at least one card from every suit, and subtract that number from the total number of possible hands.

Step-by-step explanation:

In the given card game, each player is dealt a hand consisting of 5 cards from a deck of 52 cards. The deck consists of 4 suits and 13 ranks in each suit. To find the number of ways to deal a hand that contains at least one card from every suit:

1. Calculate the total number of possible hands: This can be done by selecting any 5 cards out of 52, using the combination formula. The number of total possible hands is 52C5.
2. Calculate the number of hands that do not contain at least one card from every suit: To do this, we can subtract the number of hands with cards from only 3 suits, 2 suits, or 1 suit from the total number of possible hands.
3. The number of hands with cards from only 3 suits: There are 4 ways to choose 3 suits, and for each choice, we need to select 5 cards from the chosen suits. This can be done by multiplying 4C3 with [(13C5)^3] since we select 5 cards from each chosen suit.
4. The number of hands with cards from only 2 suits: There are 6 ways to choose 2 suits, and for each choice, we need to select 5 cards from the chosen suits. This can be done by multiplying 4C2 with [(13C5)^2] since we select 5 cards from each chosen suit.
5. The number of hands with cards from only 1 suit: There are 4 ways to choose 1 suit, and for each choice, we need to select 5 cards from the chosen suit. This can be done by multiplying 4C1 with 13C5 since we select 5 cards from the chosen suit.
6. Calculate the number of hands that contain at least one card from every suit: Subtract the number of hands that do not contain at least one card from every suit from the total possible hands.

By following these steps and performing the calculations, you can find the number of ways to deal a hand that contains at least one card from every suit.