Cards are drawn one by one without replacement from a pack of 52 cards. The probability of the 11th card drawn is first ace,is:

Options

A. `(451)/(884)`
B. `(241)/(1456)`
C. `(164)/(4165)`
D. none of these

Explanation

To find the probability that the 11th card drawn from a pack of 52 cards is the first Ace, we can follow these steps:

### Step 1: Understand the Problem
We need to ensure that the first 10 cards drawn are not Aces, and the 11th card drawn is an Ace.

### Step 2: Count the Total Cards
In a standard deck, there are 52 cards in total, which include:
- 4 Aces
- 48 non-Ace cards

### Step 3: Calculate the Ways to Choose the First 10 Cards
Since we want the first 10 cards to be non-Ace cards, we can choose these from the 48 non-Ace cards. The number of ways to choose 10 cards from 48 is given by the combination formula:
\[ \binom{48}{10} \]

### Step 4: Calculate the Ways to Choose the 11th Card
After drawing 10 non-Ace cards, we want the 11th card to be an Ace. There are 4 Aces available, so the number of ways to choose 1 Ace from 4 is:
\[ \binom{4}{1} \]

### Step 5: Calculate the Total Ways to Choose Any 11 Cards
The total number of ways to choose any 11 cards from the 52 cards is given by:
\[ \binom{52}{11} \]

### Step 6: Calculate the Probability
The probability that the 11th card is the first Ace can be calculated using the formula:
\[ P(\text{11th card is first Ace}) = \frac{\text{Ways to choose 10 non-Aces and 1 Ace}}{\text{Total ways to choose 11 cards}} \]
This can be expressed as:
\[ P = \frac{\binom{48}{10} \cdot \binom{4}{1}}{\binom{52}{11}} \]

### Step 7: Substitute the Values
Now we substitute the values into the formula:
\[ \binom{48}{10} = \frac{48!}{10!(48-10)!}, \quad \binom{4}{1} = 4, \quad \text{and} \quad \binom{52}{11} = \frac{52!}{11!(52-11)!} \]

### Step 8: Simplify the Calculation
Calculating these values:
1. Calculate \(\binom{48}{10}\)
2. Calculate \(\binom{52}{11}\)
3. Substitute these values into the probability formula.

### Step 9: Final Calculation
After calculating these combinations and substituting back, we find:
\[ P = \frac{164}{4165} \]

### Conclusion
Thus, the probability that the 11th card drawn is the first Ace is:
\[ \frac{164}{4165} \]