A poker hand consisting of 6 cards is dealt from a standard deck of 52 cards.

Find the probability that the hand contains exactly 3 face cards. Leave your answer as a reduced fraction.

To solve this problem, we'll calculate the probability that a 6-card poker hand contains exactly 3 face cards (Jacks, Queens, and Kings) from a standard deck of 52 cards.

### Step 1: Identify the total number of cards and face cards

1. A standard deck has 52 cards.
2. There are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).
3. The remaining 40 cards are non-face cards.

### Step 2: Total number of ways to select 6 cards from 52

The total number of ways to select 6 cards out of 52 is given by:

\[
\binom{52}{6}
\]

### Step 3: Select exactly 3 face cards and 3 non-face cards

We need exactly 3 face cards and 3 non-face cards in the hand.

1. The number of ways to select 3 face cards out of 12 face cards:

   \[
   \binom{12}{3}
   \]

2. The number of ways to select 3 non-face cards out of 40 non-face cards:

   \[
   \binom{40}{3}
   \]

### Step 4: Calculate the total number of ways to get the desired hand

The total number of ways to get exactly 3 face cards and 3 non-face cards is:

\[
\binom{12}{3} \times \binom{40}{3}
\]

### Step 5: Calculate the probability

The probability \(P\) of getting exactly 3 face cards in a 6-card hand is:

\[
P = \frac{\binom{12}{3} \times \binom{40}{3}}{\binom{52}{6}}
\]

### Step 6: Simplify the expressions

Let's compute these values:

- \(\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1}\)
- \(\binom{40}{3} = \frac{40 \times 39 \times 38}{3 \times 2 \times 1}\)
- \(\binom{52}{6} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47}{6 \times 5 \times 4 \times 3 \times 2 \times 1}\)

Let's calculate these values step-by-step and simplify the probability.The calculations give us the following:

- The number of ways to choose 3 face cards and 3 non-face cards: \( \binom{12}{3} \times \binom{40}{3} = 2,173,600 \).
- The total number of ways to choose 6 cards out of 52: \( \binom{52}{6} = 20,358,520 \).

Thus, the probability of getting exactly 3 face cards in a 6-card hand is:

\[
P = \frac{2,173,600}{20,358,520}
\]

Reducing this fraction:

\[
P = \frac{2,173,600}{20,358,520} = \frac{1,087}{10,179}
\]

Would you like more details or have any questions?

### Related Questions
1. What is the probability of getting exactly 4 face cards in a 6-card hand?
2. How many ways are there to get at least one face card in a 6-card hand?
3. What is the probability of getting no face cards in a 6-card hand?
4. How does the probability change if the number of cards dealt increases to 7?
5. What is the expected number of face cards in a 6-card hand?

**Tip:** Understanding how to compute combinations helps in solving many probability-related problems efficiently!

A poker hand consisting of 6 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 face cards. Leave your answer as a reduced fraction.

Calculate the probability of getting exactly 3 face cards in a 6-card poker hand dealt from a standard deck of 52 cards. This problem involves key probability and combinatorial concepts, using combinations to find the number of ways to achieve the desired outcome. Suitable for students in Grades 10-12.

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