Four cards are drawn from a full pack of cards . Find the probability that
all the four are spades , and one of them is a king , and

Answer

To find the probability that all four drawn cards are spades and one of them is a king, we can follow these steps:

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

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

### Step 2: Determine the number of favorable outcomes.
We need to find the number of ways to draw 4 spades such that one of them is the king of spades.

1. **Identify the total number of spades:** There are 13 spades in a deck.
2. **Choose the king of spades:** There is only 1 king of spades.
3. **Choose the remaining 3 spades from the other 12 spades:** We can choose 3 spades from the remaining 12 spades.

Thus, the number of favorable outcomes is:

\[ \text{Favorable outcomes} = \binom{12}{3} \times \binom{1}{1} \]

### Step 3: Calculate the combinations.
Now we will calculate the combinations:

1. Calculate \(\binom{52}{4}\):

\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} \]

2. Calculate \(\binom{12}{3}\):

\[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} \]

### Step 4: Calculate the probability.
The probability \(P\) of drawing 4 spades with one being a king is given by:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\binom{12}{3}}{\binom{52}{4}} \]

### Step 5: Substitute the values and simplify.
Substituting the values we calculated:

\[ P = \frac{\binom{12}{3}}{\binom{52}{4}} = \frac{\frac{12 \times 11 \times 10}{3 \times 2 \times 1}}{\frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1}} \]

### Step 6: Final calculation.
Now we can simplify this expression:

\[ P = \frac{12 \times 11 \times 10 \times 4}{52 \times 51 \times 50 \times 49} \]

Calculating the final values will give us the probability.