What is the number of ways of choosing 4 cards from a deck of 52 cards? In how many of these,
(iii)At least 3 are face cards.

Answer

To solve the problem of choosing 4 cards from a deck of 52 cards such that at least 3 are face cards, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the Total Number of Face Cards:**
- In a standard deck of cards, there are 3 types of face cards: Jack, Queen, and King.
- Each type has 4 cards (one for each suit: hearts, diamonds, clubs, spades).
- Therefore, the total number of face cards = 3 (types) × 4 (cards each) = 12 face cards.

2. **Determine the Number of Non-Face Cards:**
- The total number of cards in the deck is 52.
- The number of non-face cards = Total cards - Face cards = 52 - 12 = 40 non-face cards.

3. **Calculate the Number of Ways to Choose Cards:**
- We need to find the number of ways to choose 4 cards such that at least 3 are face cards. This can happen in two scenarios:
- Case 1: 3 face cards and 1 non-face card.
- Case 2: 4 face cards.

4. **Case 1: Choosing 3 Face Cards and 1 Non-Face Card:**
- The number of ways to choose 3 face cards from 12 face cards is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose 3 face cards} = \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
- The number of ways to choose 1 non-face card from 40 non-face cards:
\[ \text{Ways to choose 1 non-face card} = \binom{40}{1} = 40 \]
- Therefore, the total ways for Case 1:
\[ \text{Total for Case 1} = 220 \times 40 = 8800 \]

5. **Case 2: Choosing 4 Face Cards:**
- The number of ways to choose 4 face cards from 12 face cards:
\[ \text{Ways to choose 4 face cards} = \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

6. **Combine Both Cases:**
- The total number of ways to choose 4 cards such that at least 3 are face cards is the sum of the two cases:
\[ \text{Total ways} = \text{Total for Case 1} + \text{Total for Case 2} = 8800 + 495 = 9295 \]

### Final Answer:
The total number of ways to choose 4 cards from a deck of 52 cards such that at least 3 are face cards is **9295**.