How many different 3 card hands can be dealt from a deck of 52 cards

To find how many different 3-card hands can be dealt from a deck of 52 cards, we need to calculate the number of ways to choose 3 cards from a set of 52. Since the order in which the cards are dealt does not matter, we use the combination formula:

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

Where:
- \( n = 52 \) (total number of cards)
- \( k = 3 \) (number of cards to choose)

Now, applying the formula:

\[
\binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = \frac{52 \times 51 \times 50}{6}
\]

Let's calculate:

\[
\frac{52 \times 51 \times 50}{6} = \frac{132600}{6} = 22100
\]

So, there are **22,100 different 3-card hands** that can be dealt from a deck of 52 cards.

Would you like more details or have any questions about this? 

Here are 5 related questions to consider:
1. How many different 5-card hands can be dealt from a 52-card deck?
2. What if the order of the cards mattered—how many ways could you deal 3 cards then?
3. How does the combination formula differ from the permutation formula?
4. How many 3-card hands could be dealt from a deck missing 2 cards?
5. If there are 4 jokers added to the deck, how many 3-card hands can be dealt?

**Tip:** Always remember, combinations are used when the order doesn't matter, while permutations are used when the order does matter!

How many different 3 card hands can be dealt from a deck of 52 cards?

Discover how to calculate the number of different 3-card hands that can be dealt from a 52-card deck using the combination formula. This combinatorics problem covers essential concepts and formulas, such as C(n, k) = n! / (k!(n - k)!), making it ideal for students in Grades 10-12.

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