Four cards are drawn from a full pack of cards . Find the probability that
all are diamonds ,

Answer

To find the probability that all four cards drawn from a full pack of cards are diamonds, we can follow these steps:

### Step 1: Understand the Total Outcomes
A standard deck of cards contains 52 cards. We need to determine the total number of ways to draw 4 cards from these 52 cards. This can be calculated using the combination formula:

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

### Step 2: Calculate the Total Outcomes
Using the combination formula, we can express this as:

\[ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52!}{4! \cdot 48!} \]

### Step 3: Understand the Favorable Outcomes
Next, we need to find the number of favorable outcomes, which is the number of ways to draw 4 diamonds from the 13 diamonds available in the deck. This can also be calculated using the combination formula:

\[ \text{Favorable Outcomes} = \binom{13}{4} \]

### Step 4: Calculate the Favorable Outcomes
Using the combination formula again, we can express this as:

\[ \binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13!}{4! \cdot 9!} \]

### Step 5: Set Up the Probability Formula
The probability of drawing 4 diamonds can be calculated using the formula for probability:

\[ P(\text{All Diamonds}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{\binom{13}{4}}{\binom{52}{4}} \]

### Step 6: Substitute and Simplify
Substituting the values we calculated:

\[ P(\text{All Diamonds}) = \frac{\frac{13!}{4! \cdot 9!}}{\frac{52!}{4! \cdot 48!}} \]

The \(4!\) cancels out:

\[ P(\text{All Diamonds}) = \frac{13! \cdot 48!}{9! \cdot 52!} \]

### Step 7: Cancel the Factorials
Now we can cancel \(9!\) and \(48!\):

\[ P(\text{All Diamonds}) = \frac{13 \times 12 \times 11 \times 10}{52 \times 51 \times 50 \times 49} \]

### Step 8: Calculate the Final Probability
Now we compute the values:

\[ P(\text{All Diamonds}) = \frac{17160}{6497400} \]

### Step 9: Simplify the Fraction
On simplifying:

\[ P(\text{All Diamonds}) = \frac{11}{4165} \]

Thus, the probability that all four cards drawn are diamonds is:

\[ \boxed{\frac{11}{4165}} \]