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!