How many 10
-card hands contain four cards of the same value?

To solve the problem of determining how many 10-card hands contain four cards of the same value, let's break down the process step-by-step:

1. **Choose the value for the four cards**: There are 13 different values in a standard deck of cards (2, 3, 4, ..., 10, J, Q, K, A). We need to choose one of these values to have four cards of that value.
   \[
   \text{Number of ways to choose the value} = 13
   \]

2. **Choose 4 suits for the four cards of the chosen value**: For the chosen value, there are 4 suits and we need to choose all 4 suits (since we need exactly four cards of the same value).
   \[
   \text{Number of ways to choose 4 suits} = \binom{4}{4} = 1
   \]

3. **Choose the remaining 6 cards from the other 48 cards**: After choosing the four cards, we have 48 cards left in the deck. We need to choose 6 more cards from these 48 cards. 
   \[
   \text{Number of ways to choose 6 cards from 48} = \binom{48}{6}
   \]

Putting it all together, the total number of ways to form a 10-card hand with exactly four cards of the same value is calculated by multiplying these individual choices together:
\[
\text{Total number of ways} = 13 \times 1 \times \binom{48}{6}
\]

4. **Calculate \(\binom{48}{6}\)**: This is the number of combinations of 48 items taken 6 at a time.
   \[
   \binom{48}{6} = \frac{48!}{6!(48-6)!} = \frac{48!}{6! \cdot 42!}
   \]

Using a calculator or software to compute this,
\[
\binom{48}{6} \approx 12,271,512
\]

5. **Calculate the final number of hands**:
   \[
   13 \times 12,271,512 = 159,529,656
   \]

So, the number of 10-card hands that contain exactly four cards of the same value is **159,529,656**.

Would you like more details or have any other questions?

Here are 5 questions you might be interested in asking:
1. How do you calculate combinations and permutations?
2. Can you explain the concept of binomial coefficients?
3. How many 10-card hands contain three cards of the same value?
4. What is the probability of drawing a 10-card hand with four cards of the same value from a standard deck?
5. How does the inclusion-exclusion principle work in probability?

Tip: When dealing with combinatorics problems, breaking down the problem into smaller steps and understanding the constraints is crucial for arriving at the correct solution.

Learn how to calculate the number of 10-card hands that contain exactly four cards of the same value from a standard deck. This combinatorics problem involves choosing a card value, selecting suits, and calculating combinations. Suitable for advanced high school students and those interested in probability and combinatorics.

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