Four cards are drawn at random from a pack of 52 playing cards, one of them is King other is Queen, third is Jack and fourth is Ace. If the probability is `K/(.^(52)C_(4))`, then K is

Options

A. 64
B. 4
C. 16
D. 256

Answer

To solve the problem, we need to find the value of \( K \) in the probability expression given in the question. The probability is expressed as \( \frac{K}{\binom{52}{4}} \).

### Step-by-Step Solution:

1. **Understand the Problem**: We need to draw 4 specific cards from a deck of 52 cards: one King, one Queen, one Jack, and one Ace.

2. **Total Ways to Choose 4 Cards**: The total number of ways to choose 4 cards from 52 is given by the combination formula:
\[ \text{Total ways} = \binom{52}{4} \]

3. **Choosing the Specific Cards**:
- There are 4 Kings in the deck, and we need to choose 1. The number of ways to choose 1 King from 4 is:
\[ \binom{4}{1} = 4 \]
- Similarly, for the Queen, Jack, and Ace, the number of ways to choose each is also:
\[ \binom{4}{1} = 4 \quad \text{(for Queen)} \]
\[ \binom{4}{1} = 4 \quad \text{(for Jack)} \]
\[ \binom{4}{1} = 4 \quad \text{(for Ace)} \]

4. **Total Favorable Outcomes**: Since we need to select one of each card (King, Queen, Jack, Ace), and these selections are independent, we multiply the number of ways to choose each:
\[ \text{Favorable outcomes} = \binom{4}{1} \times \binom{4}{1} \times \binom{4}{1} \times \binom{4}{1} = 4 \times 4 \times 4 \times 4 = 4^4 \]

5. **Calculate \( 4^4 \)**:
\[ 4^4 = 256 \]

6. **Express Probability**: The probability of drawing one King, one Queen, one Jack, and one Ace is:
\[ P(\text{1 King, 1 Queen, 1 Jack, 1 Ace}) = \frac{4^4}{\binom{52}{4}} = \frac{256}{\binom{52}{4}} \]

7. **Identify \( K \)**: From the probability expression, we have:
\[ \frac{K}{\binom{52}{4}} = \frac{256}{\binom{52}{4}} \]
Hence, \( K = 256 \).

### Final Answer:
The value of \( K \) is \( 256 \).