A hand of 8 cards is dealt from a regular deck of 52. How many different hands are possible if the hand contains exactly 3 sevens?

To find the number of different hands possible where a hand of 8 cards contains exactly 3 sevens from a regular deck of 52 cards, we can approach the problem step by step:

### Step 1: Choose 3 sevens.
- A standard deck has four sevens (7♠, 7♥, 7♦, 7♣).
- We need to select 3 of these four sevens.
  
The number of ways to choose 3 sevens from 4 is given by the combination:

\[
\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4
\]

### Step 2: Choose 5 non-sevens.
- After choosing 3 sevens, there are \( 52 - 4 = 48 \) non-seven cards left in the deck.
- We need to choose 5 cards from these remaining 48 cards.
  
The number of ways to choose 5 cards from 48 is given by the combination:

\[
\binom{48}{5} = \frac{48!}{5!(48-5)!}
\]

### Step 3: Multiply the results.
- The total number of different hands is the product of the two combinations:

\[
\text{Total hands} = \binom{4}{3} \times \binom{48}{5}
\]

Substituting the values:

\[
\text{Total hands} = 4 \times \binom{48}{5} = 4 \times 1,712,304 = 6,849,216
\]

### Final Answer:
There are **6,849,216** different hands possible with exactly 3 sevens in an 8-card hand.

Would you like more details on combinations or any step here?

### Related questions:
1. How many hands are possible if the hand contains exactly 2 sevens?
2. What is the probability of getting exactly 3 sevens in an 8-card hand?
3. How many different hands are possible if the hand contains at least one seven?
4. How many different hands are possible if the hand contains exactly 4 sevens?
5. What is the probability of getting no sevens in an 8-card hand?

**Tip:** The combination formula \( \binom{n}{k} \) is useful for counting selections without regard to order.

Discover how to calculate the number of different 8-card hands containing exactly 3 sevens from a deck of 52 cards. This combinatorics problem explores choosing cards using the combination formula, with a detailed step-by-step solution. Suitable for Grades 9-12.

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