Three cards are drawn at random from a pack of 52 cards. What is the probability that the drawn cards are all kings ?

Answer

To find the probability that all three drawn cards from a pack of 52 cards are kings, we can follow these steps:

### Step 1: Determine the total number of ways to choose 3 cards from 52.
The total number of ways to choose 3 cards from a pack of 52 cards is given by the combination formula:

\[ \text{Total ways} = \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52!}{3! \cdot 49!} \]

### Step 2: Simplify the combination.
We can simplify this expression:

\[ \binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = \frac{52 \times 51 \times 50}{6} \]

### Step 3: Calculate the total ways.
Calculating the above expression:

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

So, the total number of ways to choose 3 cards from 52 is 22100.

### Step 4: Determine the number of favorable outcomes (drawing 3 kings).
There are 4 kings in a deck of cards. The number of ways to choose 3 kings from these 4 is:

\[ \text{Favorable ways} = \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = 4 \]

### Step 5: Calculate the probability.
The probability \( P \) that all three drawn cards are kings is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{3 kings}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{22100} \]

### Step 6: Simplify the probability.
We can simplify this fraction:

\[ P(\text{3 kings}) = \frac{1}{5525} \]

### Final Answer:
The probability that all three drawn cards are kings is:

\[ \boxed{\frac{1}{5525}} \]