Four cards are drawn at random from a pack of 52 cards find the probability of getting all the four cards of different numbers .

Options

A. `(141 )/( 4165 )`
B. `(117 )/(833 )`
C. `(264 )/(4165)`
D. none of these

Answer

To solve the problem of finding the probability of drawing 4 cards from a standard deck of 52 cards such that all 4 cards have different numbers, we can follow these steps:

### Step 1: Understand the Total Number of Outcomes
The total number of ways to choose 4 cards from a deck of 52 cards can be calculated using combinations. The formula for combinations is given by:

\[ \text{Total Outcomes} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For our case, we need to choose 4 cards from 52:

\[ \text{Total Outcomes} = \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

### Step 2: Calculate the Favorable Outcomes
Next, we need to find the number of ways to draw 4 cards such that all cards have different numbers. In a standard deck, there are 13 different ranks (Ace through King).

1. **Choose 4 different ranks**: The number of ways to choose 4 different ranks from 13 is given by:

\[ \text{Ways to choose ranks} = \binom{13}{4} \]

2. **Choose one card from each rank**: For each of the 4 chosen ranks, there are 4 cards (one from each suit). Therefore, for each rank chosen, we can choose 1 card in 4 ways. Since we have 4 ranks, the total number of ways to choose the cards is:

\[ \text{Ways to choose cards} = 4^4 \]

Putting it all together, the total number of favorable outcomes is:

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

Calculating this:

\[ \binom{13}{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715 \]

\[ 4^4 = 256 \]

Thus,

\[ \text{Favorable Outcomes} = 715 \times 256 = 183040 \]

### Step 3: Calculate the Probability
Now, we can find the probability of drawing 4 cards of different numbers by dividing the number of favorable outcomes by the total number of outcomes:

\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{183040}{270725} \]

### Step 4: Simplify the Probability
To simplify \( \frac{183040}{270725} \), we can use a calculator or perform long division to find the decimal or simplest fraction form.

### Final Answer
Thus, the probability of drawing 4 cards from a deck of 52 cards such that all 4 cards have different numbers is:

\[ \text{Probability} \approx 0.676 \]