A poker hand consisting of 9 cards is dealt from a standard deck of 52 cards.

Find the probability that the hand contains exactly 2 face cards. Leave your answer as a reduced fraction.

The probability is

To find the probability that a poker hand consisting of 9 cards contains exactly 2 face cards, we'll proceed step-by-step. Here's the solution:

### Step 1: Total number of face cards
In a standard deck of 52 cards, there are 3 face cards (Jack, Queen, and King) in each of the 4 suits, making a total of:
\[
3 \times 4 = 12 \text{ face cards}.
\]

### Step 2: Total number of non-face cards
The deck has 52 cards, and if 12 of them are face cards, the remaining cards are non-face cards:
\[
52 - 12 = 40 \text{ non-face cards}.
\]

### Step 3: Total number of ways to choose 9 cards from a deck of 52
The total number of ways to select 9 cards from the deck of 52 is given by the combination formula:
\[
\binom{52}{9} = \frac{52!}{9!(52-9)!}.
\]

### Step 4: Number of ways to choose exactly 2 face cards and 7 non-face cards
- The number of ways to choose exactly 2 face cards from the 12 face cards is:
\[
\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66.
\]
- The number of ways to choose 7 non-face cards from the 40 non-face cards is:
\[
\binom{40}{7} = \frac{40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 18,643,560.
\]

### Step 5: Total number of favorable outcomes
The total number of favorable outcomes, where we get exactly 2 face cards and 7 non-face cards, is the product of the two combinations:
\[
\binom{12}{2} \times \binom{40}{7} = 66 \times 18,643,560 = 1,230,477,960.
\]

### Step 6: Total number of possible outcomes
The total number of ways to choose any 9 cards from 52 cards is:
\[
\binom{52}{9} = 20,358,520,620.
\]

### Step 7: Probability calculation
The probability of getting exactly 2 face cards is the ratio of favorable outcomes to the total outcomes:
\[
\text{Probability} = \frac{1,230,477,960}{20,358,520,620}.
\]
This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). After simplifying, we get:
\[
\text{Probability} = \frac{123,047,796}{2,035,852,062}.
\]

Thus, the probability is \( \frac{123,047,796}{2,035,852,062} \).

Do you have any questions, or would you like further details on any step?

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Here are 5 related questions:
1. What is the probability of getting exactly 3 face cards in the same hand?
2. How many ways are there to deal a poker hand containing at least 2 face cards?
3. What is the probability of dealing a hand with no face cards?
4. How would the probability change if the hand had 10 cards instead of 9?
5. What is the probability of getting all 4 Kings in a 9-card poker hand?

**Tip:** In combinatorics, always double-check your combination formulas to avoid small calculation errors!

A poker hand consisting of 9 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 2 face cards. Leave your answer as a reduced fraction.

Explore how to calculate the probability of dealing exactly 2 face cards from a 9-card poker hand from a standard deck of 52 cards. This problem uses combinatorics and basic probability concepts to break down the solution step by step. Suitable for students in Grades 10-12.

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