A 13 card hand is dealt from a well-shuffled standard 52-card deck.  What is the probability that 3 red cards and 10 black cards are dealt?
 0.0227


 0.0717


 0.0217


 0.0517


 0.0817


 None of the above.


To solve this, let’s break it down step-by-step.

1. **Identify Total Possible Hands**: 
   - A 13-card hand is dealt from a standard 52-card deck.
   - The number of ways to choose any 13 cards from 52 is:
     \[
     \binom{52}{13}
     \]

2. **Specific Favorable Hands**:
   - We need exactly 3 red cards and 10 black cards in the hand.
   - There are 26 red cards and 26 black cards in the deck.

   - The number of ways to choose 3 red cards from 26 red cards is:
     \[
     \binom{26}{3}
     \]

   - The number of ways to choose 10 black cards from 26 black cards is:
     \[
     \binom{26}{10}
     \]

   - Therefore, the total number of favorable hands is:
     \[
     \binom{26}{3} \times \binom{26}{10}
     \]

3. **Calculate the Probability**:
   - The probability of getting exactly 3 red cards and 10 black cards is the ratio of the favorable hands to the total possible hands:
     \[
     \frac{\binom{26}{3} \times \binom{26}{10}}{\binom{52}{13}}
     \]

Now, let’s compute this:

- \(\binom{26}{3} = 2600\)
- \(\binom{26}{10} = 5311735\)
- \(\binom{52}{13} = 635,013,559,600\)

So,
\[
\text{Probability} = \frac{2600 \times 5311735}{635013559600} = \frac{13810411000}{635013559600} \approx 0.0217
\]

### Answer
The probability is approximately **0.0217**, so the correct answer is:
**0.0217**

Would you like more details or have any questions?

---

1. What is the probability of getting 5 red and 8 black cards?
2. How does the probability change if we draw 10 cards instead of 13?
3. What’s the probability of getting only red cards in a 13-card hand?
4. How many ways are there to draw 3 red and 10 black cards?
5. How would the solution change for a deck with jokers?

**Tip**: Using combinations simplifies probability calculations for card draws without replacement.

A 13 card hand is dealt from a well-shuffled standard 52-card deck. What is the probability that 3 red cards and 10 black cards are dealt?
0.0227
0.0717
0.0217
0.0517
0.0817
None of the above.

Calculate the probability of drawing exactly 3 red and 10 black cards from a 52-card deck in a 13-card hand. This problem covers concepts in probability and combinatorics, including the combination formula and probability calculations, suitable for students in Grades 11-12.

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