In the game of poker, determine the number of ways a straight (five cards with consecutive values, such as A 2 3 4 5 or 7 8 9 10 J or 10 J Q K A, but not all of the same suit) can be picked.
To determine the number of ways a straight can be picked in a game of poker, we need to consider a few factors:
1. Number of possible starting cards: Since a straight consists of five cards with consecutive values, we need to calculate the number of ways we can choose the first card of the straight.
2. Number of possible combinations for the remaining cards: Once we select the first card, the remaining four cards must also be in consecutive order. We need to calculate the number of possible combinations for these remaining cards.
3. Account for ACE: In a straight, an ACE can be considered both as the highest card (A, 2, 3, 4, 5) and as the lowest card (10, J, Q, K, A). We need to ensure that both possibilities are accounted for in our calculation.
Let's break down these factors one by one:
1. Number of possible starting cards:
For the first card of the straight, we have 10 options: A, 2, 3, 4, 5, 6, 7, 8, 9, 10 (as we can't use J, Q, K, or A to start the straight since that would give a royal flush or a five-of-a-kind). Therefore, we have 10 possible starting cards.
2. Number of possible combinations for the remaining cards:
Once the first card is chosen, the remaining four cards must be chosen from a reduced deck. We have excluded the values already used for the first card, but we still have all the suits available. There are 4 suits in total (hearts, diamonds, clubs, and spades). For each card in the straight, we have 4 possible suits to choose from. Therefore, for the remaining four cards, we have 4*4*4*4 = 4^4 = 256 possible combinations.
3. Account for ACE:
As mentioned earlier, an ACE can be considered both as the highest card and the lowest card in a straight. So, we need to include both possibilities.
Considering these factors, we can determine the total number of ways a straight can be picked in poker by multiplying the results:
Total number = Number of possible starting cards * Number of possible combinations for the remaining cards * Number of possibilities for ACE
Total number = 10 * 256 * 2 = 5120 ways
Therefore, there are 5120 ways a straight can be picked in a game of poker.