How many ways are there to deal hands from a standard playing deck (13 values and 4 suits) to five players if each player gets five cards and the rest of the cards remain in the deck?

To determine how many ways there are to deal hands from a standard 52-card deck to five players, where each player gets five cards and the remaining cards stay in the deck, we need to calculate the number of possible distributions of the cards.

### Step 1: Determine the total number of cards to be dealt
Each of the five players receives five cards, so:
\[
5 \times 5 = 25 \text{ cards}
\]
Thus, 25 cards are dealt out of the 52 cards in the deck.

### Step 2: Calculate the number of ways to choose 25 cards from 52
The number of ways to select 25 cards from 52 (without considering the order of selection) is given by the binomial coefficient:
\[
\binom{52}{25} = \frac{52!}{25!(52-25)!} = \frac{52!}{25! \cdot 27!}
\]
This gives the number of ways to choose 25 cards out of 52.

### Step 3: Calculate the number of ways to distribute the selected 25 cards among 5 players
Once we have selected the 25 cards, we need to distribute them to the five players. The number of ways to do this is calculated by considering that each player receives 5 cards, and the order in which they receive their cards matters. This is equivalent to dividing the 25 cards into 5 groups of 5, and the number of ways to do this is given by:
\[
\frac{25!}{(5!)^5}
\]
Here, \(25!\) accounts for all possible permutations of the 25 cards, and \((5!)^5\) corrects for the fact that the order of the cards within each player's hand doesn't matter.

### Step 4: Combine the results
The total number of ways to deal the hands is the product of the number of ways to choose 25 cards and the number of ways to distribute those 25 cards among the players:
\[
\text{Total number of ways} = \binom{52}{25} \times \frac{25!}{(5!)^5}
\]

### Final Expression:
The total number of ways to deal hands is given by:
\[
\text{Total number of ways} = \frac{52!}{25! \times 27!} \times \frac{25!}{(5!)^5} = \frac{52!}{27! \times (5!)^5}
\]

This expression represents the total number of ways to deal the hands under the given conditions.

Would you like further details or have any questions about this calculation?

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Here are five related questions to explore:

1. What if the order in which cards are dealt matters? How does that change the calculation?
2. How would the calculation differ if the number of players was different?
3. Can you calculate the total number of possible 5-card hands from a deck of 52 cards?
4. How does the probability of getting a specific 5-card hand differ from the total number of possible hands?
5. What if each player received a different number of cards? How would that affect the number of ways to deal the hands?

**Tip:** When calculating combinations or permutations, it's helpful to break the problem down into stages, as it simplifies complex calculations.

Explore how to calculate the number of ways to deal hands from a standard deck of cards to five players, where each player gets five cards. This math problem involves combinatorial concepts such as binomial coefficients and permutations.

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