QuestionFrom a standard 52-card deck, how many five-card hands consist of one card of one denomination, one card of another denomination, and three cards of a third denomination?
The number of possible hands is . (Simplify your answer.)
Studdy Solution
STEP 1
1. We are using a standard 52-card deck.
2. A five-card hand consists of cards with specific denominations.
3. The hand consists of one card of one denomination, one card of another denomination, and three cards of a third denomination.
4. We need to calculate the number of such possible hands.
STEP 2
1. Choose the denominations for the cards.
2. Select the specific cards for each chosen denomination.
3. Calculate the total number of possible hands.
STEP 3
Choose the denominations for the cards.
- First, choose the denomination for the three cards. There are 13 possible denominations, so there are 13 ways to choose this denomination.
- Next, choose the denominations for the two single cards. There are 12 remaining denominations after choosing the first, and we need to choose 2 more. This can be done in ways.
STEP 4
Select the specific cards for each chosen denomination.
- For the denomination with three cards, choose 3 cards from the 4 available in that denomination. This can be done in ways.
- For each of the two single-card denominations, choose 1 card from the 4 available. This can be done in ways for each.
STEP 5
Calculate the total number of possible hands.
- Multiply the number of ways to choose the denominations by the number of ways to choose the specific cards for each denomination:
- Simplify the expression:
- Calculate:
The number of possible hands is:
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