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.