Four cards are successive drawn without replacement from a pack of cards. What is the probability that all the four are aces ?

Answer

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

### Step 1: Understand the total number of cards and aces
In a standard deck of cards, there are a total of 52 cards, out of which 4 are aces.

### Step 2: Determine the total ways to draw 4 cards
Since we are drawing cards without replacement, the total number of ways to choose 4 cards from 52 is given by the combination formula \( \binom{n}{r} \), which is calculated as:

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

### Step 3: Calculate the number of favorable outcomes
The number of ways to choose 4 aces from the 4 available aces is:

\[ \binom{4}{4} = 1 \]

### Step 4: Calculate the probability
The probability \( P \) that all four drawn cards are aces is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(\text{4 aces}) = \frac{\text{Number of ways to choose 4 aces}}{\text{Total ways to choose 4 cards}} = \frac{\binom{4}{4}}{\binom{52}{4}} = \frac{1}{\binom{52}{4}} \]

### Step 5: Substitute the combination value
Now, we need to calculate \( \binom{52}{4} \):

\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = \frac{6497400}{24} = 270725 \]

### Step 6: Final probability calculation
Now substituting back into the probability formula:

\[ P(\text{4 aces}) = \frac{1}{270725} \]

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

\[ \boxed{\frac{1}{270725}} \]

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