(i)What is the number of ways of choosing 4 cards from a deck of 52 cards?
(ii) In how many of these 3 are red and 1 is black

Answer

To solve the problem step by step, we will break it down into two parts as stated in the question.

### Part (i): Number of ways to choose 4 cards from a deck of 52 cards

1. **Understanding the Problem**: We need to choose 4 cards from a total of 52 cards. Since the order in which we choose the cards does not matter, we will use combinations.

2. **Using the Combination Formula**: The formula for combinations is given by:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

3. **Applying the Formula**: Here, \( n = 52 \) and \( r = 4 \). Therefore, we need to calculate \( 52C4 \):
\[ 52C4 = \frac{52!}{4!(52-4)!} = \frac{52!}{4! \cdot 48!} \]

4. **Calculating the Values**: We can simplify this as follows:
\[ 52C4 = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} \]

5. **Final Calculation**:
- Calculate the numerator: \( 52 \times 51 \times 50 \times 49 = 6497400 \)
- Calculate the denominator: \( 4 \times 3 \times 2 \times 1 = 24 \)
- Now divide:
\[ 52C4 = \frac{6497400}{24} = 270725 \]

Thus, the number of ways to choose 4 cards from a deck of 52 cards is **270725**.

### Part (ii): Number of ways to choose 3 red cards and 1 black card

1. **Understanding the Composition of the Deck**: In a standard deck of 52 cards, there are 26 red cards and 26 black cards.

2. **Choosing the Red Cards**: We need to choose 3 red cards from the 26 red cards. We will use the combination formula again:
\[ 26C3 = \frac{26!}{3!(26-3)!} = \frac{26!}{3! \cdot 23!} \]

3. **Calculating \( 26C3 \)**:
\[ 26C3 = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} \]
- Calculate the numerator: \( 26 \times 25 \times 24 = 15600 \)
- Calculate the denominator: \( 3 \times 2 \times 1 = 6 \)
- Now divide:
\[ 26C3 = \frac{15600}{6} = 2600 \]

4. **Choosing the Black Card**: Now we need to choose 1 black card from the 26 black cards:
\[ 26C1 = \frac{26!}{1!(26-1)!} = 26 \]

5. **Combining the Choices**: Since we need to choose both 3 red cards and 1 black card, we multiply the number of ways:
\[ \text{Total ways} = 26C3 \times 26C1 = 2600 \times 26 = 67600 \]

Thus, the number of ways to choose 3 red cards and 1 black card is **67600**.

### Summary of Solutions:
- (i) The number of ways to choose 4 cards from a deck of 52 cards is **270725**.
- (ii) The number of ways to choose 3 red cards and 1 black card is **67600**.