find the number of possible 5 card hands that contain the cards specified
a: 5 face cards(either king,queen, or jacks)
b: 4 aces and 1 other card
c: 2 aces and 3 kings
please explain what im supposed to do thanks!
792
sorry bro dont know
To find the number of possible 5-card hands that contain the specified cards, we need to consider the principles of combinations and counting.
a) To find the number of 5-card hands that contain 5 face cards (king, queen, or jack), we first need to determine the total number of face cards in a standard deck. Since there are 4 kings, 4 queens, and 4 jacks, there are a total of 12 face cards.
Next, we'll use the concept of combinations. For this case, we need to choose 5 face cards from a pool of 12 face cards. The order in which the cards are selected doesn't matter, so we use the combination formula:
C(n, r) = n! / (r!(n - r)!),
where n is the number of items to choose from and r is the number of items to be chosen.
Using this formula, the number of possible 5-card hands containing face cards only is:
C(12, 5) = 12! / (5!(12 - 5)!)
= 792.
Therefore, there are 792 possible 5-card hands containing any combination of kings, queens, or jacks.
b) To find the number of 5-card hands that contain 4 aces and 1 other card, we need to consider the total number of aces in a standard deck, which is 4.
First, we choose 4 aces from the available 4 aces. This can be done in only one way. Next, we pick 1 card from the remaining 48 cards (52 cards total minus 4 aces).
Therefore, the number of possible 5-card hands containing 4 aces and 1 other card is:
1 * C(48, 1) = 48.
Hence, there are 48 possible 5-card hands with these specifications.
c) To find the number of 5-card hands that contain 2 aces and 3 kings, we need to consider the total number of aces (4) and kings (4) in a standard deck.
First, we choose 2 aces from 4 aces, which can be done in C(4, 2) = 6 ways. Next, we choose 3 kings from 4 kings, which can be done in C(4, 3) = 4 ways. Finally, we multiply the number of possibilities for choosing aces and kings together:
C(4, 2) * C(4, 3) = 6 * 4 = 24.
Hence, there are 24 possible 5-card hands containing 2 aces and 3 kings.
To summarize:
a) There are 792 possible 5-card hands containing 5 face cards.
b) There are 48 possible 5-card hands containing 4 aces and 1 other card.
c) There are 24 possible 5-card hands containing 2 aces and 3 kings.