Four cards are drawn at random from a pack of 52 playing cards , Find the probability of getting
all the four cards of the same number ,

Answer

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

### Step 1: Calculate the total number of ways to select four cards from 52 cards.
The total number of ways to choose 4 cards from 52 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total ways} = \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} \]

Calculating this gives:

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

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

Thus, the number of favorable outcomes is:

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

### Step 3: Calculate the probability.
The probability \( P \) of drawing four cards of the same number is given by the ratio of the number of favorable outcomes to the total number of outcomes.

\[ P(\text{all four cards of the same number}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{13}{270725} \]

### Final Answer:
Thus, the probability of drawing four cards of the same number is:

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

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