How many poker hands of five cards are dealt from a deck of 52 cards? Also, how many ways are there to select 47 cards from a standard deck of 52 cards?

To solve these problems, we will use combinatorics, specifically the concept of combinations, where the order of selection does not matter.

### 1. Number of poker hands (5 cards) dealt from a deck of 52 cards:
This is a combination problem where you are selecting 5 cards from a deck of 52. The formula for combinations is:

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

Where:
- \( n = 52 \) (the total number of cards),
- \( k = 5 \) (the number of cards in a poker hand).

So,

\[
C(52, 5) = \frac{52!}{5!(52-5)!} = \frac{52!}{5! \cdot 47!}
\]

We can calculate this using factorial values:

\[
C(52, 5) = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960
\]

So, there are **2,598,960** different ways to deal a 5-card poker hand from a deck of 52 cards.

### 2. Number of ways to select 47 cards from a deck of 52:
This is another combination problem, but this time we are selecting 47 cards from 52. The formula is:

\[
C(52, 47) = \frac{52!}{47!(52-47)!} = \frac{52!}{47! \cdot 5!}
\]

Notice that this is the same calculation as selecting 5 cards from 52, because:

\[
C(52, 47) = C(52, 5)
\]

Therefore, the number of ways to select 47 cards from a deck of 52 is also **2,598,960**.

### Conclusion:
- There are **2,598,960** ways to deal a 5-card poker hand from a deck of 52 cards.
- There are also **2,598,960** ways to select 47 cards from a deck of 52.

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Would you like more details on any of these calculations? 

Here are 5 related questions for further exploration:
1. How many ways can a 7-card hand be dealt from a 52-card deck?
2. What is the probability of being dealt a flush (all cards of the same suit) in a 5-card poker hand?
3. How many different 2-card combinations can be made from a deck of 52 cards?
4. How many ways can you select 3 red cards from a deck of 26 red cards?
5. What is the probability of being dealt a straight in a 5-card poker hand?

**Tip:** Combinations are a key concept in probability and counting problems where the order does not matter, whereas permutations are used when the order does matter.

Discover how to calculate the number of possible 5-card poker hands that can be dealt from a 52-card deck. Learn about combinations and the use of factorials with step-by-step explanations, suitable for high school students studying combinatorics.

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