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

What is the probability you are dealt exactly 2 heart(s)? Enter your answer as an unreduced fraction.


To calculate the probability of being dealt exactly 2 hearts in a 5-card hand from a standard deck of 52 cards, we follow these steps:

1. **Determine the number of ways to choose 2 hearts:**
   There are 13 hearts in the deck. The number of ways to choose 2 hearts from these 13 is given by the combination formula:

   \[
   \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78
   \]

2. **Determine the number of ways to choose 3 non-hearts:**
   There are 39 non-heart cards (52 total cards - 13 hearts). The number of ways to choose 3 non-heart cards from these 39 is:

   \[
   \binom{39}{3} = \frac{39!}{3!(39-3)!} = \frac{39 \times 38 \times 37}{3 \times 2 \times 1} = 9139
   \]

3. **Calculate the total number of ways to choose 5 cards from the deck:**
   The total number of ways to choose 5 cards from 52 is:

   \[
   \binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960
   \]

4. **Calculate the total number of ways to get exactly 2 hearts:**
   Multiply the number of ways to choose the hearts by the number of ways to choose the non-hearts:

   \[
   \text{Ways to get exactly 2 hearts} = \binom{13}{2} \times \binom{39}{3} = 78 \times 9139 = 714282
   \]

5. **Calculate the probability:**
   Finally, the probability of being dealt exactly 2 hearts is the number of favorable outcomes divided by the total number of outcomes:

   \[
   P(\text{exactly 2 hearts}) = \frac{\text{Ways to get exactly 2 hearts}}{\text{Total ways to choose 5 cards}} = \frac{714282}{2598960}
   \]

Thus, the probability of being dealt exactly 2 hearts is:

\[
\frac{714282}{2598960}
\]

This fraction can be simplified if necessary, but as per your request, it is presented in unreduced form. 

Would you like further details on any part of this calculation? Here are some related questions you might consider:

1. What are the different types of combinations and their applications?
2. How does the concept of probability apply to card games?
3. What is the formula for calculating combinations?
4. Can you explain the significance of the total number of cards in a deck?
5. How would the probability change if we were looking for 3 hearts instead? 

**Tip:** Always double-check calculations for accuracy, especially when working with combinations and large numbers!

A 5-card hand is dealt from a standard deck of cards. What is the probability you are dealt exactly 2 heart(s)? Enter your answer as an unreduced fraction.

Learn how to calculate the probability of being dealt exactly 2 hearts in a 5-card hand from a standard 52-card deck. This detailed solution explores concepts of probability and combinations, using key formulas to find the number of ways to select hearts and non-hearts. Suitable for students in Grades 10-12.

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