From a standard 52-card deck, how many 7-card hands consist of 3 hearts and 4 diamonds? From a standard 52-card deck, how many 4-card hands contain a card from each suit?
The correct answer and explanation is :
1. Number of 7-card hands consisting of 3 hearts and 4 diamonds:
A standard deck of 52 cards consists of four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. To find how many 7-card hands consist of exactly 3 hearts and 4 diamonds, we need to calculate the following:
- Step 1: Choosing 3 hearts: There are 13 hearts in the deck, and we want to choose 3. The number of ways to choose 3 hearts from 13 is given by the combination formula: [
\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286
] - Step 2: Choosing 4 diamonds: Similarly, there are 13 diamonds, and we want to choose 4. The number of ways to choose 4 diamonds from 13 is: [
\binom{13}{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715
] - Step 3: Total number of hands: To find the total number of 7-card hands with 3 hearts and 4 diamonds, we multiply the number of ways to choose the hearts by the number of ways to choose the diamonds: [
\binom{13}{3} \times \binom{13}{4} = 286 \times 715 = 204,590
]
Thus, the total number of 7-card hands consisting of 3 hearts and 4 diamonds is 204,590.
2. Number of 4-card hands containing a card from each suit:
In a standard deck, we have 4 suits: hearts, diamonds, clubs, and spades, each containing 13 cards. We want to select 4 cards such that one card comes from each suit.
- Step 1: Choosing 1 card from each suit: For each suit, we have 13 cards, so the number of ways to select 1 card from each suit is: [
\binom{13}{1} = 13 \quad \text{for each suit}
] - Step 2: Total number of ways: Since we are choosing 1 card from each suit, the total number of ways to form the hand is the product of the combinations from each suit: [
13 \times 13 \times 13 \times 13 = 13^4 = 28,561
]
Thus, the total number of 4-card hands containing one card from each suit is 28,561.
Explanation:
Both problems rely on the use of combinations because the order of the cards in the hand does not matter. In the first problem, we are specifically choosing a set number of cards from certain suits (hearts and diamonds), and the combination formula is used to calculate the number of ways to select those cards. In the second problem, we must choose one card from each suit, and we calculate the total number of ways to do that by multiplying the number of choices for each suit.