A 5-card hand is dealt from a standard deck of cards.

What is the probability you are dealt exactly 3 heart(s)? Enter your answer as an unreduced fraction.

To calculate the probability of being dealt exactly 3 hearts in a 5-card hand from a standard deck, we’ll use combinatorics. Here's the step-by-step process:

1. **Determine the total number of possible 5-card hands**:
   There are \(52\) cards in a deck, so the number of ways to choose any 5 cards from 52 is:
   \[
   \binom{52}{5} = \frac{52!}{5!(52 - 5)!} = \binom{52}{5}
   \]

2. **Calculate the number of ways to get exactly 3 hearts**:
   - There are 13 hearts in a deck, and we need exactly 3 of them, so the number of ways to choose 3 hearts from 13 is:
     \[
     \binom{13}{3} = \frac{13!}{3!(13 - 3)!} = \binom{13}{3}
     \]
   - The remaining 2 cards must not be hearts. There are \(52 - 13 = 39\) non-heart cards, so the number of ways to choose 2 non-hearts from 39 is:
     \[
     \binom{39}{2} = \frac{39!}{2!(39 - 2)!} = \binom{39}{2}
     \]

3. **Combine these to find the total number of successful outcomes**:
   The number of ways to get exactly 3 hearts and 2 non-hearts is:
   \[
   \binom{13}{3} \times \binom{39}{2}
   \]

4. **Calculate the probability**:
   The probability is the number of successful outcomes divided by the total number of 5-card hands:
   \[
   \frac{\binom{13}{3} \times \binom{39}{2}}{\binom{52}{5}}
   \]

Now, we substitute in the values:

- \(\binom{13}{3} = 286\)
- \(\binom{39}{2} = 741\)
- \(\binom{52}{5} = 2,598,960\)

So the probability is:
\[
\frac{286 \times 741}{2,598,960} = \frac{211926}{2598960}
\]

Thus, the probability of being dealt exactly 3 hearts is:
\[
\frac{211926}{2598960}
\]

Would you like further details on any of these steps?

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### Expanding Questions

1. What is the probability of getting exactly 4 hearts in a 5-card hand?
2. How would the probability change if we wanted exactly 2 hearts instead of 3?
3. How does the probability calculation change if we are only drawing 4 cards?
4. What is the probability of drawing a 5-card hand with no hearts at all?
5. How many ways are there to get exactly 3 diamonds in a 5-card hand?

**Tip:** When calculating combinations, always check if values can be simplified before multiplying, as it can make computations easier.

A 5-card hand is dealt from a standard deck of cards. What is the probability you are dealt exactly 3 heart(s)? Enter your answer as an unreduced fraction.

Learn how to calculate the probability of being dealt exactly 3 hearts in a 5-card hand from a standard deck. The solution uses combinatorics to find the number of successful outcomes and applies the probability formula for card hands. The result is expressed as an unreduced fraction.

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