Four cards are drawn at random from a pack of 52 playing cards , Find the probability of getting
all face cards , (king , Queen , jack )

Answer

To find the probability of drawing all face cards (Kings, Queens, and Jacks) when four cards are drawn from a standard deck of 52 playing cards, we can follow these steps:

### Step 1: Determine the total number of ways to choose 4 cards from 52 cards.
The total number of ways to choose 4 cards from a deck of 52 is given by the combination formula:

\[ \text{Total ways} = \binom{52}{4} \]

Calculating this:

\[ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

### Step 2: Determine the number of favorable outcomes (selecting 4 face cards).
In a deck of cards, there are 12 face cards (3 face cards: King, Queen, Jack from each of the 4 suits). We need to find the number of ways to choose 4 face cards from these 12 face cards:

\[ \text{Favorable outcomes} = \binom{12}{4} \]

Calculating this:

\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

### Step 3: Calculate the probability of drawing all face cards.
The probability \( P \) of drawing 4 face cards is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{all face cards}) = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{\binom{12}{4}}{\binom{52}{4}} = \frac{495}{270725} \]

### Step 4: Simplify the probability.
Now, we can simplify the fraction:

\[ P(\text{all face cards}) = \frac{495}{270725} \approx 0.00183 \]

### Final Answer:
Thus, the probability of drawing all face cards when four cards are drawn from a pack of 52 playing cards is approximately:

\[ P \approx 0.00183 \]

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