To find the probability of drawing a king, a queen, and a jack from a standard deck of 52 playing cards, we can follow these steps:
### Step 1: Determine the total number of ways to draw 3 cards from 52 cards.
The total number of ways to choose 3 cards from 52 is given by the combination formula:
\[
\text{Total ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!}
\]
Where \( n \) is the total number of items (52 cards) and \( r \) is the number of items to choose (3 cards).
\[
\text{Total ways} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100
\]
### Step 2: Determine the number of favorable outcomes.
To draw a king, a queen, and a jack, we can choose one card from each of these ranks.
- There are 4 kings in the deck.
- There are 4 queens in the deck.
- There are 4 jacks in the deck.
The number of ways to choose one king, one queen, and one jack is:
\[
\text{Favorable outcomes} = 4 \times 4 \times 4 = 64
\]
### Step 3: Calculate the probability.
The probability \( P \) of drawing a king, a queen, and a jack is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[
P(\text{king, queen, jack}) = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{64}{22100}
\]
### Step 4: Simplify the probability.
To simplify the fraction, we can divide the numerator and the denominator by their greatest common divisor (GCD). In this case, the fraction is already in its simplest form.
### Final Answer:
Thus, the probability of drawing a king, a queen, and a jack from a pack of 52 playing cards is:
\[
P(\text{king, queen, jack}) = \frac{64}{22100} \approx 0.0029
\]
