From a pack of 52 playing cards, three cards are drawn at random. Find the probability of drawing a king, a queen and a jack.

Answer

To find the probability of drawing a king, a queen, and a jack from a pack of 52 playing cards, we can follow these steps:

### Step 1: Determine the Total Number of Events
The total number of ways to draw 3 cards from a deck of 52 cards can be calculated using the combination formula \( nCr \), which is given by:

\[ \text{Total Events} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} \]

### Step 2: Calculate the Total Number of Events
Calculating this gives:

\[ \binom{52}{3} = \frac{52 \times 51 \times 50}{6} = \frac{132600}{6} = 22100 \]

### Step 3: Determine the Favorable Events
Next, we need to find the number of favorable outcomes for drawing one king, one queen, and one jack.

- There are 4 kings in a deck, so the number of ways to choose 1 king is \( \binom{4}{1} = 4 \).
- There are 4 queens in a deck, so the number of ways to choose 1 queen is \( \binom{4}{1} = 4 \).
- There are 4 jacks in a deck, so the number of ways to choose 1 jack is \( \binom{4}{1} = 4 \).

Thus, the total number of favorable outcomes is:

\[ \text{Favorable Events} = 4 \times 4 \times 4 = 64 \]

### Step 4: Calculate the Probability
The probability of drawing one king, one queen, and one jack is given by the ratio of the number of favorable events to the total number of events:

\[ P(\text{King, Queen, Jack}) = \frac{\text{Favorable Events}}{\text{Total Events}} = \frac{64}{22100} \]

### Step 5: Simplify the Probability
To simplify \( \frac{64}{22100} \), we can check for common factors. The greatest common divisor (GCD) of 64 and 22100 is 4. Dividing both the numerator and denominator by 4 gives:

\[ P(\text{King, Queen, Jack}) = \frac{16}{5525} \]

### Final Answer
Thus, the probability of drawing a king, a queen, and a jack from a pack of 52 playing cards is:

\[ \frac{16}{5525} \]

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