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

Options

A. `(17 )/( 1665)`
B. `(1)/( 20825 )`
C. `(7)/( 25850 )`
D. none of these

Answer

To find the probability of drawing four cards of the same number from a standard deck of 52 cards, we can follow these steps:

### Step 1: Determine the total number of ways to choose 4 cards from 52 cards.
The total number of ways to choose 4 cards from 52 can be calculated using the combination formula:

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

Here, \( n = 52 \) and \( r = 4 \):

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

### Step 2: Calculate the total combinations.
Calculating the above expression:

\[ \text{Total combinations} = \frac{52 \times 51 \times 50 \times 49}{24} = 270725 \]

### Step 3: Determine the number of favorable outcomes.
To have all four cards of the same number, we can choose any of the 13 ranks (Ace, 2, 3, ..., 10, Jack, Queen, King). For each rank, there is only one way to choose all four cards (since there are exactly four cards of each rank in a standard deck).

Thus, the number of favorable outcomes is:

\[ \text{Favorable outcomes} = 13 \]

### Step 4: Calculate the probability.
The probability \( P \) of drawing four cards of the same number is given by the formula:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{13}{270725} \]

### Step 5: Simplify the probability.
The final probability can be expressed as:

\[ P = \frac{13}{270725} \approx 0.000048 \]

### Conclusion
Thus, the probability of drawing four cards of the same number from a deck of 52 cards is:

\[ P \approx 0.000048 \]

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