 from a standard 52 card deck, how many 7 card hands consist of 3 face cards and 4 nonface cards

To solve the problem of finding how many 7-card hands consist of 3 face cards and 4 non-face cards from a standard deck of 52 cards, we need to break it down step by step.

### Step 1: Identify Face Cards and Non-Face Cards
A standard deck has:
- 52 cards in total.
- 12 face cards (3 face cards: Jack, Queen, and King in each of the 4 suits).
- 40 non-face cards (the remaining cards from Ace through 10 across the 4 suits).

### Step 2: Choosing the Face Cards
We are required to choose 3 face cards out of the 12 available face cards. The number of ways to choose 3 face cards is calculated using combinations (denoted as \( C(n, k) \), where \( n \) is the total number and \( k \) is the number to be chosen):

\[
\text{Number of ways to choose 3 face cards} = C(12, 3) = \frac{12!}{3!(12-3)!} = \binom{12}{3}
\]

### Step 3: Choosing the Non-Face Cards
We also need to choose 4 non-face cards out of the 40 available non-face cards. Again, this is calculated using combinations:

\[
\text{Number of ways to choose 4 non-face cards} = C(40, 4) = \frac{40!}{4!(40-4)!} = \binom{40}{4}
\]

### Step 4: Multiply the Combinations
To find the total number of 7-card hands consisting of 3 face cards and 4 non-face cards, we multiply the two combinations:

\[
\text{Total number of hands} = \binom{12}{3} \times \binom{40}{4}
\]

Let's calculate these combinations.import math

# Calculate combinations for face cards and non-face cards
face_cards_combinations = math.comb(12, 3)
non_face_cards_combinations = math.comb(40, 4)

# Total number of possible 7-card hands
total_hands = face_cards_combinations * non_face_cards_combinations
total_handsThe total number of 7-card hands that consist of 3 face cards and 4 non-face cards is 20,105,800.

Would you like more details on the process or have any questions?

Here are some related questions:
1. How do you calculate combinations in general?
2. How would the calculation change if we wanted 4 face cards and 3 non-face cards?
3. What is the probability of drawing exactly 3 face cards in a 7-card hand?
4. How does this calculation differ if using a deck with jokers?
5. What are the odds of drawing a hand with no face cards at all?
6. How do you calculate the number of face cards in multiple decks?
7. How many ways can you draw 7 cards with exactly 2 face cards?
8. What if the deck contained fewer face cards?

**Tip:** Understanding combinations is key in problems where order doesn't matter—make sure to practice calculating them for various scenarios!

Learn how to calculate the number of 7-card hands consisting of 3 face cards and 4 non-face cards from a standard 52-card deck. This problem involves combinatorics and probability, specifically focusing on combinations. Suitable for advanced high school students and beyond.

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