To calculate the probability of being dealt exactly 5 face cards from a standard 52-card deck, let's break it down:

Step-by-Step Explanation:

1. Total number of cards in a deck: A standard deck has 52 cards, and there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).

2. Combinations to choose face cards: We want to choose 5 face cards out of the 12 available. The number of ways to do this is given by the combination formula: $\binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12!}{5!7!} = 792$

3. Combinations to choose non-face cards: We also need to choose the remaining 2 cards from the 40 non-face cards in the deck (52 total cards minus the 12 face cards). The number of ways to do this is: $\binom{40}{2} = \frac{40!}{2!(40-2)!} = \frac{40!}{2!38!} = 780$

4. Total possible combinations: The total number of ways to deal 7 cards from the 52-card deck (without regard to face cards) is: $\binom{52}{7} = \frac{52!}{7!(52-7)!} = \frac{52!}{7!45!} = 133,784,560$

5. Probability: The probability of being dealt exactly 5 face cards is the ratio of the favorable outcomes (choosing 5 face cards and 2 non-face cards) to the total possible outcomes (choosing 7 cards from 52): $P(\text{5 face cards}) = \frac{\binom{12}{5} \times \binom{40}{2}}{\binom{52}{7}} = \frac{792 \times 780}{133,784,560} = \frac{617,760}{133,784,560} \approx 0.004617$

Final Answer:

The probability of being dealt exactly 5 face cards is approximately 0.0046.

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Would you like any further details or clarification?

Here are 5 follow-up questions for exploration:

1. What is the probability of being dealt exactly 6 face cards?
2. How many combinations exist for being dealt no face cards at all?
3. What is the probability of being dealt exactly 4 face cards?
4. How does the probability change if the number of cards dealt is increased to 10?
5. What is the probability of being dealt at least one face card in a 7-card hand?

Tip: When calculating probabilities, always check if you're dealing with combinations (order doesn't matter) or permutations (order matters).