As an expert in card games and combinatorics, I can provide a detailed explanation of how to determine the number of 7s in a standard deck of cards. A standard deck of 52 cards is composed of four suits: clubs, diamonds, hearts, and spades, with each suit containing 13 cards. These cards are numbered from Ace (which can be considered as 1) through to 10, and then there are the face cards: Jack, Queen, and King. Therefore, within each suit, there is one card with the value of 7.

To calculate the total number of 7s in the deck, we simply multiply the number of suits by the number of 7s in each suit:

\[ \text{Total number of 7s} = \text{Number of suits} \times \text{Number of 7s per suit} \]

Given that there are 4 suits and 1 seven in each suit:

\[ \text{Total number of 7s} = 4 \times 1 = 4 \]

So, there are 4 sevens in a standard deck of cards.

Now, let's address the question about the number of 7-card hands that consist of 3 hearts and 4 diamonds. This is a combinatorial problem that can be solved using the concept of combinations, which are a way of selecting items from a larger set where the order of selection does not matter.

First, we need to calculate the number of ways to choose 3 hearts from the 13 available in the hearts suit. The formula for combinations is:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Where \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and \( ! \) denotes factorial.

For the hearts:

\[ C(13, 3) = \frac{13!}{3!(13-3)!} \]

For the diamonds:

\[ C(13, 4) = \frac{13!}{4!(13-4)!} \]

To find the total number of 7-card hands with 3 hearts and 4 diamonds, we multiply the combinations of choosing 3 hearts by the combinations of choosing 4 diamonds:

\[ \text{Total combinations} = C(13, 3) \times C(13, 4) \]

Calculating these values will give us the number of ways to form such hands.

Now, let's move on to the translation of the answer into Chinese.

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From a standard 52-card deck, how many 7-card hands consist of 3 hearts and 4 diamonds? A deck of 52 cards consists of 4 suits, namely clubs, diamonds, hearts and spades, each suit having 13 cards. The 13 cards are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jake, king and Queen.read more >>