Question

imagine you were given a standard deck of 52 cards. you are dealt four cards face-down and one card face-up. the face-up card is the ace of this partial information that you have about your hand, how many different five card hands could you have been dealt such that you now have four cards of the same rank?

Answer 1

Considering one ace is already face-up, there are 48 different five-card hands possible that contain four aces, as three aces need to be selected from the remaining three and one other non-ace card from the rest of the deck.

Step-by-step explanation:

The question asks how many different five-card hands could be created from a standard deck of 52 cards with four cards of the same rank if one ace is already dealt face-up. In a standard deck, there are four suits (clubs, diamonds, hearts, and spades) with 13 cards in each suit, including ranks from ace through king.

To create a hand with four cards of the same rank (in this case, an ace since one is already face-up), we need to select the remaining three aces from the deck. Since there are four aces in total and one is already face up, there are C(3,3) ways to pick the remaining aces.

For the fifth card, which cannot be an ace to maintain a four-of-a-kind hand, we have 48 remaining cards (52 cards minus four aces). There are 48 ways to select this single card.

Therefore, the total number of different five-card hands we could have been dealt that contain four aces, with one ace already face up, is C(3,3) × 48 = 1 × 48 = 48.