If you are dealt 

88 

cards from a shuffled deck of 52​ cards, find the probability of getting 

fourfour 

queens and 

fourfour 

kings.

We are tasked with finding the probability of being dealt 88 cards from a shuffled deck of 52 cards, and specifically receiving **four queens** and **four kings**.

### Step-by-Step Breakdown

1. **Total number of ways to choose 8 cards from 52 cards**:  
   The number of ways to choose any 8 cards from the deck is given by the combination formula:

   \[
   \binom{52}{8} = \frac{52!}{8!(52 - 8)!}
   \]

2. **Number of ways to choose 4 queens from 4 queens**:  
   There are exactly 4 queens in the deck, and we need to choose all 4. This can only be done in one way:

   \[
   \binom{4}{4} = 1
   \]

3. **Number of ways to choose 4 kings from 4 kings**:  
   Similarly, there are exactly 4 kings in the deck, and we need to choose all 4. This can also be done in one way:

   \[
   \binom{4}{4} = 1
   \]

4. **Number of ways to choose the remaining 0 cards from the other 44 cards**:  
   Since we've already selected 4 queens and 4 kings, we don't need to choose any additional cards from the remaining 44 cards (52 cards minus the 8 kings and queens). The number of ways to do this is:

   \[
   \binom{44}{0} = 1
   \]

5. **Putting it all together**:
   The number of favorable outcomes is:

   \[
   \binom{4}{4} \times \binom{4}{4} \times \binom{44}{0} = 1 \times 1 \times 1 = 1
   \]

   So, there is only **1 favorable way** to choose 4 queens and 4 kings.

6. **Total possible outcomes**:
   The total number of possible ways to choose 8 cards from 52 cards is:

   \[
   \binom{52}{8} = \frac{52!}{8!(52 - 8)!} = 752,875,620
   \]

7. **Probability**:
   The probability is the ratio of favorable outcomes to the total outcomes:

   \[
   P = \frac{1}{752,875,620}
   \]

   Therefore, the probability of being dealt 4 queens and 4 kings from 8 cards is:

   \[
   P \approx 1.33 \times 10^{-9}
   \]

### Conclusion:
The probability of being dealt 4 queens and 4 kings from a deck of 52 cards when selecting 8 cards is approximately \(1.33 \times 10^{-9}\).

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Do you need more details or have any questions?

Here are 5 related questions:

1. What is the probability of getting 4 aces and 4 kings instead?
2. How does the probability change if you are dealt 9 cards instead of 8?
3. How many ways are there to choose 2 queens and 2 kings from a deck?
4. What is the probability of getting at least 1 queen from 8 cards?
5. How do probabilities change when using a deck with jokers?

**Tip:** When calculating probabilities involving combinations, remember that symmetry in card suits often simplifies the calculations.

If you are dealt 8 cards from a shuffled deck of 52 cards, find the probability of getting four queens and four kings.

Discover how to calculate the probability of being dealt four queens and four kings from a shuffled deck of 52 cards when choosing 8 cards. Learn the step-by-step breakdown of combinatorics and probability theory, and understand the formula for calculating possible outcomes. This problem is ideal for Grades 10-12 students.

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