Four cards are drawn from a well-shuffled pack of 52 cards. The probability of obtaininge Windows 3 diamonds and one spade is

Options

A. `(""^26C_4)/(""^52C_4)`
B. `(""^(26)C_2xx""^(26)C_2)/(""^(52C_1))`
C. `(""^(13)C_3xx""^(13)C_1)/(""^(52C_4))`
D. `(""^(13)C_3xx""^(10)C_1)/(""^(52C_4))`

Answer

To find the probability of drawing 3 diamonds and 1 spade from a well-shuffled pack of 52 cards, we can follow these steps:

### Step 1: Understand the Total Number of Cards
A standard deck has 52 cards, which includes:
- 13 Diamonds
- 13 Spades
- 13 Clubs
- 13 Hearts

### Step 2: Calculate the Total Ways to Draw 4 Cards
The total number of ways to draw 4 cards from 52 cards is given by the combination formula \( \binom{n}{r} \), which is calculated as:
\[ \text{Total ways} = \binom{52}{4} \]

### Step 3: Calculate the Ways to Draw 3 Diamonds
The number of ways to choose 3 diamonds from 13 diamonds is:
\[ \text{Ways to choose 3 diamonds} = \binom{13}{3} \]

### Step 4: Calculate the Ways to Draw 1 Spade
The number of ways to choose 1 spade from 13 spades is:
\[ \text{Ways to choose 1 spade} = \binom{13}{1} \]

### Step 5: Calculate the Total Ways to Draw 3 Diamonds and 1 Spade
The total number of ways to draw 3 diamonds and 1 spade is the product of the number of ways to choose the diamonds and the spade:
\[ \text{Total ways for 3 diamonds and 1 spade} = \binom{13}{3} \times \binom{13}{1} \]

### Step 6: Calculate the Probability
The probability of drawing 3 diamonds and 1 spade is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{3 diamonds and 1 spade}) = \frac{\binom{13}{3} \times \binom{13}{1}}{\binom{52}{4}} \]

### Step 7: Substitute Values and Simplify
Now we can substitute the values into the combinations:
- \( \binom{13}{3} = \frac{13!}{3!(13-3)!} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \)
- \( \binom{13}{1} = 13 \)
- \( \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \)

Now substituting these values:
\[ P(\text{3 diamonds and 1 spade}) = \frac{286 \times 13}{270725} = \frac{3718}{270725} \]

### Final Answer
Thus, the probability of obtaining 3 diamonds and 1 spade when drawing 4 cards from a deck of 52 cards is:
\[ P(\text{3 diamonds and 1 spade}) = \frac{3718}{270725} \]