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 kind (one pair of each of two different face values and a card of a third face value)
To find the number of ways a three-of-a-kind hand can be dealt in poker, we need to consider the following:
1. Choose the two different face values for the pairs: There are 13 possible face values (Ace, 2, 3, ..., 10, Jack, Queen, King) in a deck of cards. We need to choose two face values out of these 13.
To calculate this, we use the combination formula: C(n, r) = n! / (r!(n-r)!)
In this case, we want to choose 2 face values out of 13, so the calculation is: C(13, 2) = 13! / (2!(13-2)!) = 78.
2. Choose the two suits for the first pair: For each of the two chosen face values, we need to choose two suits from the deck of four suits (Spades, Hearts, Diamonds, Clubs).
Again, we use the combination formula: C(n, r) = n! / (r!(n-r)!)
In this case, we want to choose 2 suits out of 4, so the calculation is: C(4, 2) = 4! / (2!(4-2)!) = 6.
3. Choose the suits for the third card: Once we have determined the two face values for the pairs, we have one remaining face value for the third card. We can choose any suit for this third card.
Since there are 4 suits, there are 4 possible choices for the suit of the third card.
To find the total number of ways a three-of-a-kind hand can be dealt, we multiply the number of choices for each step:
Total number of ways = (Number of face value pairs) x (Number of suit choices for the pairs) x (Number of suit choices for the third card)
Total number of ways = (78) x (6) x (4) = 1,872.
Therefore, there are 1,872 different ways to be dealt a three-of-a-kind hand in poker.