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The problem you're working on is as follows:

**You select 7 cards randomly from a deck of 52 cards. What is the probability that all of the cards selected are face cards (i.e., jacks, queens, or kings)?**

Let's break this down step by step:

### Step 1: Count of face cards
A deck of 52 cards has:
- 4 Jacks,
- 4 Queens,
- 4 Kings.
So, there are a total of \( 4 + 4 + 4 = 12 \) face cards in the deck.

### Step 2: Total number of possible selections
When you select 7 cards from a deck of 52, the total number of possible outcomes (combinations) is given by the binomial coefficient:
\[
\binom{52}{7} = \frac{52!}{7!(52 - 7)!} = \frac{52!}{7!45!}
\]

### Step 3: Number of favorable outcomes
The number of favorable outcomes, where all 7 cards are face cards, is given by the number of ways to choose 7 cards from the 12 available face cards:
\[
\binom{12}{7} = \frac{12!}{7!(12 - 7)!} = \frac{12!}{7!5!}
\]

### Step 4: Probability
The probability of selecting 7 face cards is the ratio of the number of favorable outcomes to the total number of possible outcomes:
\[
P(\text{7 face cards}) = \frac{\binom{12}{7}}{\binom{52}{7}}
\]

Now, let’s calculate these values. I'll do the computation next.The probability that all 7 cards selected are face cards (jacks, queens, or kings) is approximately \( 5.92 \times 10^{-6} \), or 0.00000592.

This is a very small probability, indicating that it's highly unlikely to randomly select 7 face cards from a deck of 52 cards.

Do you have any questions or would you like further details?

Here are 5 related questions to expand on this concept:
1. How would the probability change if you were selecting 5 cards instead of 7?
2. What is the probability of selecting at least 1 face card in 7 cards?
3. How does the number of non-face cards affect the probability?
4. What is the probability of selecting exactly 3 face cards in 7 cards?
5. How would the problem change if the deck was shuffled after each selection?

**Tip:** For probability problems involving combinations, remember that the order of selection does not matter, which is why we use the binomial coefficient formula.

You select 7 cards randomly from a deck of 52 cards. What is the probability that all of the cards selected are face cards (i.e., jacks, queens, or kings)?

Learn how to calculate the probability of selecting 7 face cards (jacks, queens, or kings) randomly from a deck of 52 cards. This solution uses concepts from combinatorics and probability, including the binomial coefficient formula. Suitable for Grades 11-12 or college-level finite mathematics students.

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