In playing poker, the likelihood of being dealt a particular hand depends on the number of ways in which that hand could have been dealt from the 52-card deck. Find the number of ways each hand could be dealt.
a) three of a
To find the number of ways a specific hand could be dealt in poker, you need to consider the combinations and permutations involved. Let's calculate the number of ways to be dealt each hand:
a) Three of a Kind:
To form a three of a kind hand, you need to choose one rank out of the thirteen available ranks in a standard 52-card deck. Once you choose the rank, you need to select three cards from the four cards of that rank. Additionally, you need to select any two other cards from the remaining 48 cards in the deck (since you don't want a full house or four of a kind).
The calculations can be summarized as follows:
Number of ways to choose the rank: C(13, 1) = 13
Number of ways to choose the three cards of that rank: C(4, 3) = 4
Number of ways to choose the two other cards: C(48, 2) = 48! / (2! * (48-2)!) = 1128
Now, we can multiply these numbers to find the total number of ways to be dealt a three of a kind hand:
Total ways = 13 * 4 * 1128 = 58,368 ways
Therefore, there are 58,368 ways to be dealt a three of a kind hand in poker from a standard 52-card deck.
To find the number of ways a three of a kind hand could be dealt, we need to consider the following:
1. Choose the rank of the three cards: There are 13 ranks in a deck (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A). Choose one of these ranks.
2. Choose the suits for the three cards: Each rank has four possible suits (clubs, diamonds, hearts, spades), so we choose 3 suits out of the 4 available.
3. Choose the remaining two cards: After choosing the rank and suits for the three of a kind, we have 2 cards left to choose. We can choose any two from the remaining 49 cards in the deck.
To calculate the number of ways, we multiply the number of choices for each step:
Number of ways = 13 * C(4, 3) * C(49, 2)
Where C(n, r) represents the combination formula which calculates the number of ways to choose r items from a set of n items.
Calculating the value:
Number of ways = 13 * 4 * C(49, 2)
To find C(49, 2), we use the formula: C(n, r) = n! / (r! * (n-r)!)
C(49, 2) = 49! / (2! * (49-2)!)
= 49! / (2! * 47!)
= (49 * 48) / (2 * 1)
= 1176
Substituting back in:
Number of ways = 13 * 4 * 1176
= 61,776
Therefore, there are 61,776 different ways to be dealt a three of a kind hand in poker.